tV THE
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Name of Book and Volume,
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187
ELEMENTS
OF
PHYSICAL MANIPULATION.
BY
EDWARD C. PICKERING,
T/iayer Professor of Physics in the Massachusetts Institute of Technology
NEW YORK:
PUBLISHED BY HURD AND HOUGHTON.
El)c ftfterfftrc
1873.
Entered according to Act of Congress in the year 1873, by
EDWARD C. PICKERING,
In the Oflice of the Librarian of Congress, Washington.
TO
A
|r4
THE FIRST TO PROPOSE A PHYSICAL LABORATORY,
THIS WORK IS MOST AFFECTIONATELY INSCRIBED
BY HIS
SINCERE FRIEND AND PUPIL,
THE AUTHOR.
PREFACE.
THE rapid spread of the Laboratory System of teaching Physics,
both in this country and abroad, seems to render imperative the
demand for a special text-book, to be used by the student. To
meet this want the present work has been prepared, based on
the experience gained in the Massachusetts Institute of Technol-
ogy during the past four years. The preliminary chapter is
devoted to general methods of investigation, and the more com-
mon applications of the mathematics to the discussion of results.
The graphical method does not seem to have attracted the atten-
tion it deserves ; it is accordingly compared here with the
analytical method. Some new developments of it are moreover
inserted. It is of fundamental importance that the student should
clearly understand how to deal with his observations, and reduce
them, and that he should be familiar with the various kinds of
errors present in all physical experiments. A short description is
also given of the various methods of measuring distances, time
and weights, which, in fact, form the basis of all physical inves-
tigation. This chapter is intended as the ground-work of a short
course of lectures, given to the students before they begin their
work in the laboratory. It should be so far extended by the
instructor, as 10 render them familiar with the general principles
on which all physical instruments are constructed, thus greatly
(v)
VI PREFACE.
aiding them when they have occasion to devise apparatus for
their own work.
The remainder of the volume is devoted to a series of experi-
ments which it is intended that the student shall perform in the
laboratory. Each experiment is divided into two parts ; the first
called Apparatus, giving a description of the instruments re-
quired, and designed to aid the instructor in preparing the labora-
tory for the class. The student should read this over, and with it
the second part, entitled Experiment, which explains in detail
what he is to do.
Perhaps the greatest advantage to be derived from a course of
physical manipulation, is the means it affords of teaching a student
to think for himself. This should be encouraged by allowing him
to carry out any ideas that may occur to him, and so far as possible
devise and construct, with his own hands, the apparatus needed.
Many such investigations are suggested in connection with some of
the experiments, for instance Nos. 13, 37, 48, 69, 77, 93 and others.
To aid in this work, a room adjoining the laboratory should be fit-
ted up with a lathe and tools for working in metals and wood, as
most excellent results may sometimes be attained at very small
expense, by apparatus thus constructed by students.
The method of conducting a Physical Laboratory, for which this
book is especially designed, and which has been in daily use with
entire success at the Institute, is as follows. Each experiment is
assigned to a table, on which the necessary apparatus is kept, and
where it is always used. A board called an indicator is hung on
the wall of the room, and carries two sets of cards opposite each
other, one bearing the names of the experiments, the other those
of the students. When the class enters the laboratory, each
member goes to the indicator, sees what experiment is assigned
to him, then to the proper table where he finds the instruments
required, and by the aid of the book performs the experiment.
PREFACE. Vli
Any additional directions needed are written on a card also placed
on the table. As soon as the experiment is completed, he reports
the results to the instructor, who furnishes him with a piece of
paper divided into squares if a curve is to be constructed, or with
a blank to be filled out, when single measurements only have been
taken. In either case a blank form is supplied, as a copy. New
work is then assigned to him by merely moving his card opposite
any unoccupied experiment. By following this plan an instructor
can readily superintend classes of about twenty at a time, and is
free to pass continually from one to another, answering questions
and seeing that no mistakes are made. He can also select such ex-
periments as are suited to the requirements or ability of each stu-
dent, the order in which they are performed being of little impor-
tance, as the class is supposed to have previously attained a
moderate familiarity with the general principles of physics. More-
over, the apparatus never being moved, the danger of injury or
breakage is thus greatly lessened and much time is saved. To
avoid delay, the number of experiments ready at any time should
be greater than that of the students, and the easier ones should be
gradually replaced by those of greater difficulty.
Among these experiments several novelties, here published for
the first time, have been introduced. For instance, the apparatus
for ruling scales, p. 59, the photometers, pp. 132 and 134, and the
polarimeter, p. 221. It is also believed that the directions for
weighing, p. 47, and the adjustments for the optical circle, p. 142, if
not new, at least present the subject in a more concise and practi-
cal form than that commonly given. In fact it has been the
object throughout to give definite directions, so far as possible, as
if addressing the student in person. English weights and measures
are occasionally used as well as French to familiarize the student
with both systems, as in many of the practical applications of phys-
ics the general prevalence of the foot and pound as units seems
Vlll PREFACE.
to render premature the exclusive introduction of the metric sys-
tem. The second volume of this work, including Heat, Electric-
ity, a list of books of reference, and other matters of general
interest to the physicist, will be issued at as early a date as pos-
sible.
It is difficult to give credit for all the aid rendered in preparing
this work, as the author has for years made it a practice to collect
for it information from all available sources. He is much indebted
to Mr. Alvan Clark for the method of testing telescope lenses, and
to Prof. F. E. Stimpson for advice and aid on photometry and
other matters. The course in photography is essentially that
given by Mr. Whipple to the students at the Institute. His
especial thanks are due to his friend Prof. Cross, whose careful
examination of the proof sheets, and whose excellent judgment
has been of great assistance. Finally, if this volume, notwith-
standing its shortcomings, aids in any way those engaged in
physical investigations, either the student in the laboratory or the
amateur experimenter, the object of the author will have been
accomplished.
E. C. P.
April 29^, 1873.
INDEX,
GENERAL METHODS OF PHYSICAL INVESTIGATION.
ANALYTICAL METHOD. ......... 3
Mean, 3. Probable Error, 3. Weights, 4. Probable Error of
Two or More Variables, 4. Peirce's Criterion, 6. Differences, 6.
Interpolation, 7. Inverse Interpolation, 8. Numerical Computa-
tion, 9. Significant Figures, 10. Successive Approximations, 10.
GRAPHICAL METHOD ......... 11
Interpolation, 12. Residual Curves, 12. Maxima and Minima, IS.
Points of Inflexion, 13. Asymptotes, 14. Curves of Error, 14.
TAree Variables, 14.
PHYSICAL MEASUREMENTS 16
Time, 16. Weight, 19. Length, 19. ^Ireas, 22. Volumes, 22.
Angles, 23. Curvature, 25.
GENERAL EXPERIMENTS.
1. ESTIMATION OF TENTHS 27
2. VERNIERS 28
3. INSERTION OF CROSS-HAIRS ' . 29
4. SUSPENSION BY SILK FIBRES 31
5. TEMPERATURE CURVE .31
6. TESTING THERMOMETERS 32
7. ECCENTRICITY OF GRADUATED CIRCLES .... 33
8. CONTOUR LINES 34
9. CLEANING MERCURY ........ 35
10. CALIBRATION BY MERCURY 37
(ix)
X INDEX.
11. CALIBRATION BY WATER . 39
12. CATHETOMETER 39
13. HOOK GAUGE 41
14. SPHEROMETER • . .42
15. ESTIMATION OF TENTHS OF A SECOND .... 44
16. RATING CHRONOMETERS 44
17. MAKING WEIGHTS 46
Proper Method of Weighing . . . . . . .47
18. DECANTING GASES 50
Reduction of Gases to Standard Temperature and Pressure . 5 1
19. STANDARDS OF VOLUME 52
20. READING MICROSCOPES 55
21. DIVIDING ENGINE 56
22. RULING SCALES 59
MECHANICS OF SOLIDS.
23. COMPOSITION OF FORCES 62
24. MOMENTS 63
25. PARALLEL FORCES 64
26. CENTRE OF GRAVITY 66
27. CATENARY . 67
28. CRANK MOTION .68
29. HOOK'S UNIVERSAL JOINT 69
30. COEFFICIENT OF FRICTION 70
31. ANGLE OF FRICTION 71
32. BREAKING WEIGHT . ....... 72
33. LAWS OF TENSION 73
34. CHANGE OF VOLUME BY TENSION 75
35. DEFLECTION OF BEAMS. 1 77
36. DEFLECTION OF BEAMS. H. 79
37. TRUSSES 80
38. LAWS OF TORSION 82
39. FALLING BODIES 84
40. METRONOME PENDULUM 85
41. BORDA'S PENDULUM 85
42. TORSION PENDULUM 87
MECHANICS OF LIQUIDS AND GASES.
43. PRINCIPLES OF ARCHIMEDES 89
44. RELATIONS OF WEIGHTS AND MEASURES 90
INDEX. Xi
45. HYDROMETERS 91
46. SPECIFIC GRAVITY BOTTLE 92
47. HYDROSTATIC BALANCE . . . . . . .93
48. EFFLUX OF LIQUIDS 94
49. JETS OF WATER 97
50. RESISTANCE OF PIPES 98
51. FLOW OF LIQUIDS THROUGH SMALL ORIFICES . . .99
52. CAPILLARITY 100
53. PLATEAU'S EXPERIMENT 101
54. PNEUMATICS •••'••.. 103
55. MARIOTTE'S LAW 107
56. GAS-HOLDER 109
57. GAS-METERS Ill
58. BAROMETER 114
Measurement of Heights by the Barometer . . . .116
59. BUNSEN PUMP 118
60. AIR-METER 120
SOUND.
61. SIRENE 122
62. KUNDT'S EXPERIMENT . . . . . . . .123
63. MELDE'S EXPERIMENT 124
64. ACOUSTIC CURVES . 125
65. LISSAJOUS' EXPERIMENT .128
66. CHLADNI'S EXPERIMENT 130
LIGHT.
67. PHOTOMETER FOR ABSORPTION . . . . . 132
68. DAYLIGHT PHOTOMETER 134
69. BUNSEN PHOTOMETER 135
70. LAW OF REFLECTION 138
71. ANGLES OF CRYSTALS 139
72. ANGLE OF PRISMS 141
73. LAW OF REFRACTION. 1 145
74. LAW OF REFRACTION. II 146
75. INDEX OF REFRACTION . . . . . . . . 147
76. CHEMICAL SPECTROSCOPE 148
77. SOLAR SPECTROSCOPE ........ 151
78. LAW OF LENSES 155
79. MICROSCOPE 156
80. PREPARATION OF OBJECTS . . . • . . .167
Xll INDEX.
81. MOUNTING OBJECTS . . . . • . . .170
82. Foci AND APERTURE OF OBJECTIVES 173
83. TESTING PLANE SURFACES 175
84. TESTING TELESCOPES 178
85. PHOTOGRAPHY I. GLASS NEGATIVES 181
86. PHOTOGRAPHY II. PAPER POSITIVES 187
87. TESTING THE EYE 191
88. OPTHALMOSCOPE . . . . . . . . .196
89. INTERFERENCE OF LIGHT 199
90. DIFFRACTION 202
91. WAVE LENGTHS .205
92. POLARIZED LIGHT . . . 208
93. POLARISCOPE 217
94. SACCHARIMETER . . . .... . . . 222
GENERAL METHODS
OP
PHYSICAL INVESTIGATION.
THE object of all Physical Investigation is to determine the
effects of certain natural forces, such as gravity, cohesion, heat,
light and electricity. For this purpose we subject various bodies
to the action of these forces, and note under what circumstances
the desired effect is produced ; this is called an experiment. In-
vestigations may be of several kinds. First, we may simply wish
to know whether a certain effect can be produced, and if so, what
are the necessary conditions. To take a familiar example, we find
that water when heated boils, and that this result is attained
whether the heat is caused by burning coal, wood or gas, or by
concentrating the sun's rays ; also whether the water is contained
in a vessel of metal or glass, and finally that the same effect
may be produced with almost all other liquids. Such work is
called Qualitative, since no measurements are needed, but only to
determine the quality or kind of conditions necessary for its fulfil-
ment. Secondly, we may wish to know the magnitude of the force
required, or the temperature necessary to produce ebullition.
This we should find to be about 100° C. or 212° F., but varying
slightly with the nature of the vessel and the pressure of the air.
Thirdly, we often find two quantities so related that any change
in one produces a corresponding change in the other, and we may
wish to find the law by which we can compute the second, having
given any value of the first. Thus by changing the pressure to
which the water is subjected, we may alter the temperature of
boiling, and to determine the law by which these two quantities
are connected, hundreds of experiments have been made by physi-
2 ERRORS.
cists in all parts of the world. The last two classes of experi-
ments are called Quantitative, since accurate measurements must
be made of the quantity or magnitude of the forces involved.
Most of the following experiments are of this nature, since they
require more skill in their performance, and we can test with more,
certainty how accurately they have been done. Having obtained
a number of measurements, we next proceed to discuss them by
the aid of the mathematical principles described below, and finally
to draw our conclusions from them. It is by this method that the
whole science of Physics has been built up step by step.
Errors. In comparing a number of measurements of the same
quantity, we always find that they differ slightly from one another,
however carefully they may be made, owing to the imperfection of
all human instruments, and of our own senses. These deviations
or errors must not be confounded with mistakes, or observations
where a number is recorded incorrectly, or the experiment improp-
erly performed; such results must be entirely rejected, and not
taken into consideration in drawing our conclusions.
If we knew the true value, and subtracted it from each of our
measurements, the differences would be the errors, and these may
be divided into two kinds. We have first, constant errors, such as
a wrong length of our scale, incorrect rate of our clock, or natural
tendency of the observer to always estimate certain quantities too
great, and others too small. When we change our variables
these errors often alter also, but generally according to some defi-
nite law. When they alternately increase and diminish the result
at regular intervals they are called periodic errors. If we know
their magnitude they do no harm, since we can allow for them,
and thus obtain a value as accurate as if they did not exist. The
second class of errors are those which are due to looseness of the
joints of our instruments, impossibility of reading very small dis-
tances by the eye, &c., which sometimes render the result too large,
sometimes too small. They are called accidental errors, and are
unavoidable ; they must be carefully distinguished from the mis-
takes referred to above.
Analytical and Graphical Methods. There are two ways of
discussing the results of our experiments mathematically. By the
first, or Analytical Method, we represent each quantity by a letter,
ANALYTICAL METHOD. 3
and then by means of algebraic methods and the calculus draw
our conclusions. By the Graphical Method quantities are repre-
sented by lines or distances, and are then treated geometrically.
The former method is the most accurate, and would generally
be the best, were it not for the accidental errors, and were all
physical laws represented by simple equations. The Graphical
Method has, however, the advantage of quickness, and of enabling
us to see at a glance the accuracy of our results.
ANALYTICAL METHOD.
Mean. Suppose we have a number of observations, Al , A2 , A3 ,
A± , <fcc., differing from one another only by the accidental errors,
and we wish to find what value A is most likely to be correct. If
A was the true value, A± — A, A2 — A, &c., would be the errors
of each observation, and it is proved by the Theory of Probabil-
ities that the most probable value of A is that which makes the
sum of the squares of the errors a minimum. Also that this prop-
erty is possessed by the arithmetical mean. Hence, when we
have n such observations, we take A = (A1-}-A2-\-A8-\- &c.) -r- w,
or divide their sum by n. Thus the mean of 32, 33, 31, 30, 34, is
160 -7- 5 = 32. It is often more convenient to subtract some even
number from all the observations, and add it to the mean of the re-
mainder; thus, to find the mean of 1582, 1581, 1583, 1581, 1582,
subtract 1580 from each, and we have the remainders 2, 1, 3, 1, 2.
Their mean is 9 -f- 5 = 1.8, which added to 1580 gives 1581.8.
Where many numbers are to be added, Webb's Adder may be
used with advantage.
Probable Error. Having by the method just given, found the
most probable value of A, we next wish to know how much reli-
ance we may place on it. If it is just an even chance that the
true value is greater or less than A by E, then E is called its
probable error. To find this quantity, subtract the mean from
each of the observed values, and place A1 — A = el , A2 — A
= 02, &c. Now the theory of probabilities shows that E =
.67-v/e!2 + e22 + &c., -r- n, from which we can compute E in any
special case. As an example, suppose we have measured the
height of the barometer twenty-five times, and find the mean
29.526 with a probable error of .001 inches. Then it is an even
4 PROBABLE ERROR.
chance that the true reading is more than 29.525, and less than
29.527. Now let us suppose that some other day we make a single
reading, and wish to know its probable error. The theory of
probabilities shows that the accuracy is proportional to the square
root of the number of observations, or that the mean of four, is
only twice as accurate as a single reading, the mean of a hundred,
ten times as accurate as one. Hence in our example we have
1 : -v/25 = .001 : .005, the probable error of a single reading.
Substituting in the formula, we have the probable error of a single
reading, E' = E X Jn — &lje? + e/ + <fcc. 4- \/n. It is gene-
rally best to compute Er as well as E, and thus learn how much
dependence can be placed on a single reading of our instrument.
Weights. We have assumed in the above paragraph that all
our observations are subject to the same errors, and hence are
equally reliable. Frequently various methods are used to obtain
the same result, and some being more accurate than others are said
to have greater weight. Again, if one was obtained as the mean
of two, and the second of three similar observations, their weights
would be proportional to these numbers, and the simplest way to
allow for the weights of observations is to assume that each is
duplicated a number of times proportional to its weight. From
this statement it evidently follows that instead of the mean of a
series of measurements, we should multiply each by its weight,
and divide by the sum of the weights. Calling A1 , A2 , &c., the
measurements, and tol5 w2, &c., their weights, the best value to use
will be A = (A1wi -{- Azw2-\- &c.) -f- (w± + w2 + <fcc.). We may
always compute the weight of a series of n observations, if we
know the errors el , e2 , &c., using the formula w = n 4- 2(e^ + e22 +
e* -f- &c.). Substituting this value in the equation for probable error,
we deduce E = All 4- +/nw if all the observations have the
same weight, or -E'— .477 -f- ^/w^ + w2 + &c., if their weights are
Wl , W2 , &C.
Probable Error of Two or More Variables. Suppose we have
a number of observations of several quantities, aj, y, 2, and know
that they are so connected that we shall always have 0 = 1 -{- ax
+ by + cz. If the first term of the equation does not equal 1, we
may make it so, by dividing each term by it. Call the various
values x assumes #', a", #'", those of y, y', y", y"', and those
TWO OR MORE VARIABLES. 5
of 2, 2', 2", 2"', and so on for any other variables which may enter.
If we have more observations than variables, it will not in general
be possible to find any values of a, b and c which will satisfy them
all, but we shall always find the left hand side of our equation
instead of being zero will become some small quantity, e'^ e"^ e"\ so
that we shall have : —
e' = l+axf +V + cz',
e" = 1 + ax" + ly" + cz",
e'" =1 + axf" + by"' + cz"',
and so on, one equation corresponding to each observation. These
are called equations of condition. Now we wish to know what
are the most probable values of a, b and c, that is, those which will
make the errors er, e", e'", as small as possible. As before, we must
have the sum of the squares of the errors a minimum. We there-
fore square each equation of condition, and take their sum ; differ-
entiate this with regard to a, b and c, successively, and place each
differential coefficient equal to zero. These last are called normal
equations, and correspond to each of the quantities a, b and c, re-
spectively. The practical rule for obtaining the normal equations
is as follows : — Multiply each equation of condition by its value
of x (or coefficient of a), take their sum and equate it to zero.
Thus xr(l -f axf -[- by' + czr) + x"(l + ax" + by" + cz") + &c.
= 0, is the first normal equation. Do the same with regard to y,
and each other variable in turn. We thus obtain as many equations
as there are quantities a, b and c to be determined. Solving them
with regard to these last quantities, and substituting in the original
formula 0 = 1 + ax + by -f- cz, we have the desired equation.
As an example, suppose we have the three points, Fig. 1, whose
coordinates are xr = l,y' = 1, x" = 2, y" = 2, of" = 3, yr" = 4,
and we wish to pass a straight line as nearly as possible through
them all. We have for our equations of condi-
tions : 0 = 1 + a + 5, 0 = 1 + 2a + 25, 0 =
1 + 3a + 4&. Applying our rule, we multiply
the first equation by 1, the second by 2, and the
third by 3, the three values of a, and take their
sum, which gives l + #~{~#-|-2-|-4:a-|-4£_|_3 ^
+ 9a + l'2b — 6 + 14a + 17b = 0. For our sec- Fig.
ond normal equation we multiply by 1, 2 and 4,
6 PEIRCE'S CRITERION.
respectively, and obtain in the same way 7 + 17a + 215 = 0.
Solving, we find a — — 1.4, b = .8, and substituting in our original
equation 0 = 1 + ax -j- by, we bave 0 = 1 — \Ax + .8y, or y =
1.75aj — 1.25. Constructing tbe line tbus found, we obtain MN,
Fig. 1, which will be seen to agree very well with our original
conditions.
For a fuller description of the various applications of the Theory
of Probabilities to the discussion of observations, the reader is re-
ferred to the following works. Methode des Moindres Carrees par
Ch. Fr. Gauss, trad, par J. Bertrand, Paris, 1855, Watson's
Astronomy, 360, Chauvenefs Astronomy, II, 500, and Todhunt-
er's History of the Theory of Probabilities. A good brief de-
scription is given in Darned and Peck's Math. Diet., 454, 536 arid
590, also in Mayer's Lecture Notes on Physics, 29.
Peirce's Criterion. It has already been stated that all observa-
tions affected by errors not accidental, or mistakes, should be at
once rejected. But it is generally difficult to detect them, and
hence various Criteria have been suggested to enable us to decide
whether to reject an observation which appears to differ consid-
erably from the rest. One of the best known of these is Peirce's
Criterion, which may be defined as follows : — The proposed ob-
servations should be rejected when the probability of the system
of errors obtained by retaining them is less than that of the system
of errors obtained by their rejection, multiplied by the probability
of making so many and no more abnormal observations. Or, to
put it in a simpler but less accurate form, reject any observations
which increase the probable error, allowing for the chances of
making so many and no more erroneous measurements. Without
this last clause we might reject all but one, when the probable
error by the formula would become zero. See Gould's Astron.
Journ., 1852, II, 161; IV, 81, 137, 145.
Another criterion has been proposed by Chauvenet, which,
though less accurate than the above, is much more easily applied.
It is fully described in Watson's Astronomy, 410.
Differences. To determine the law by which a change in any
quantity A alters a second quantity B, we frequently measure B
wnen A is allowed to alter continually by equal amounts. Thus
in the example of the boiling of water, we measure the pressure
INTERPOLATION.
A. B.
D'.
D". D'".
10
cprresponding to temperatures of 0°, 10°, 20°, 30°, &c. "Writing
these numbers in a table, by placing the various values of A in
the first column, those of B in
the second, we form a third
column, in which each term is
found by subtracting the value
of B from that preceding it;
the remainders are called the
first differences Dr. In the
same way we obtain the sec-
ond differences U"t by sub-
tracting each first difference
from that which follows it, and so on.
Interpolation. One of the most common applications of differ-
ences is to determine the value of B for any intermediate value
of A. This is done by the formula,
0° 4.6
+ 4.6
9.2 -f 3.6
-f- 8.2 -j- 2.3
20° 17.4 4- 5.9
4- 14.1 -f- 3.4
30° 31.5 -f 9.3
+ 23.4
40° 54.9
A. B.
in which Bm is the measurement next preceding
./>„/", the 1st, 2d, 3d, differences, and n a fraction equal to (A —
^4.m) -f- (Am+1 — Am), in which A, Am correspond to B, -Bm, and
Am+l is the next term of the
series to Am. The use of this
formula is best shown by an
example. Suppose, from the
accompanying table, we wish
to find the value of B corre-
sponding to A = 12.5. We
have Bm = 1728, Dm' = 469,
2>m" = 78, Dm'" =G,Am =
12, Am + l = 13, A = 12.5
and n = (12.5 — 12) -f- (13
— 12) = .5. Hence,
B = 1728 + .5(469) + ^=^(78) +
10 1000
-f 331
11 1331 4- 66
-f 397 -f 6
12 1728 +72 0
' 4-469 4-6
13 2197 4-78 0
4-547 '4-6
14 2744 -|-84 0
4-631 4-6
15 3375 4- 90
4- 721
16 4096
+ 0,
B = 1728 4- 234.5 — 9.75 4- .375 + 0 = 1953.125.
In this particular case B is always the cube of A, and it may be
$ INVERSE INTERPOLATION.
seen that our formula gives an exact result. The reason is that
the 4th, and all following differences, equal zero.
Inverse Interpolation. Next suppose that in the above example
we desired the value of A for some given value of .#, as B'\ that
is, in the equation,
m
1. 12 1. 2*. o
+ &c.
we wish to find n. Evidently it is impossible to determine this
exactly, but an 'approximate value may be found by the method
of successive corrections. Neglect all terms after the third, and
deduce n from the equation,
which is a simple quadratic equation. Substitute this value of n
in the terms we have neglected, and call the result -ZVJ then
from which again we may deduce a more accurate valne of n.
This again gives a new value of JVJ and by continuing this process
we finally deduce n with any required degree of accuracy.
It is sometimes more convenient to neglect the third term, and
deduce n from the equation IB' = JBm + n^mi which saves solv-
ing a quadratic equation, but requires more approximations. The
values of n(n — 1) -f-1. 2, n(n — 1) (n — 2) -f- 1. 2. 3, &c., may
be more readily obtained from Interpolation tables than by compu-
tation. A good explanation of this subject is given in the Assu-
rance Magazine, XI, 61, XI, 301, and XII, 136, by Woolhouse.
When the terms are not equidistant the method of interpolation
by differences cannot be applied. In this case, if we wish to find
values of IB corresponding to known values of A, we assume the
equation, IB = a + bA + cAz -f- dA3 + &c., and see what values
of a, 5, c, &c., will best satisfy these equations. If we have a
great many corresponding values of A and j5, the method of least
squares should be applied. In general, however, it is much more
convenient to solve this problem by the Graphical Method de-
NUMERICAL COMPUTATION.
9
scribed below. See Cauchy's Calculus, I, 513, and an article in
the Connaissance des Temps, for 1852, by Villarceau.
Numerical Computation. Where much arithmetical work is
necessary to reduce a series of observations, a great saving of time
is effected by making the computation in a systematic form. In
general, measurements of the same quantity should be written in a
column, one below the other, instead of on the same line, and
plenty of room should always be allowed on the paper. When
the same computations must be made for several values of one of
the variables, instead of completing one before beginning the
next, it is better to carry all on together, as in the following
example. Suppose, as in the experiment of the Universal Joint,
we wish to compute the values of b in the formula, tan b = cos A.
ta;n a, in which A = 45°, and a in turn 5°, 10°, 15°, &c. Con-
struct a table thus : —
15°
25°
30°
log tan a
8.94195
9.24632
9.42805
9.56107
9.66867
9.76144
log cos A
9.84948
9.84948
9.84948
9.84948
9.84948
9.84948
log tan b
8.79143
9.09580
9.27753
9.41055
9.41815
9.61092
b
3° 32'
7° 6'
10° 44'
14° 26'
14° 41'
15'
In the first line write the various values of a, in the second the
corresponding values of its log tan, and so on throughout the com-
putation. An error is purposely committed in the above table to
show how easily it may be detected. It will be noticed that
the values of b increase pretty regularly, except that when a — 25°,
and that this is but little greater than that corresponding to a =
20°. Following the column up we find that the same is the case for
log tan b but not for log tan a, hence the error is between the two.
In fact, in the addition of the logarithms we took 6 and 4 equal
to 10, and omitted to carry the 1 ; log tan b then really equals
9.51815, and b = 18° 15'. If the error is not found at once this
value of b should be recomputed. Besides these advantages, this
method is much quicker and less laborious. When we have to
multiply, or divide by, the same number A a great many times, it
is often shorter to obtain at once 1^4, 2.4, 3.4, 4^4, &c., and use
these numbers instead of making the multiplication each time.
This is useful in reducing metres to inches, &c. There are many
10 SIGNIFICANT FIGURES.
other arithmetical devices, but their consideration would lead us
too far from our subject.
Significant Figures. One of the most common mistakes in
reducing observations is to retain more decimal places than the
experiment warrants. For instance, suppose we are measuring a
distance with a scale of millimetres, and dividing them into tenths
by the eye, we find it 32.7 mm. Now to reduce it to inches we
have 1 metre = 39.37 in., hence 32.7 mm. = 1.287399. But it is
absurd to retain the last three figures, since in our original meas-
urement, as we only read to tenths of a millimetre, we are always
liable to an error of one half this amount, or .002 of an inch.
Then we merely know that our distance lies between 1.2894 and
1.2854 inches, showing that even the thousandths are doubtful. It
is worse than useless to retain more figures, since they might mis-
lead a reader by making him think greater accuracy of measure-
ment had been attained.
If we are sure that our errors do not exceed one per cent, of the
quantity measured, we say that we have two significant figures, if
one tenth of a per cent, three, if one hundredth, four. Thus in the
example given above, if we are sure the distance is nearer 32.7
than 32.8 or 32.6, we have three significant figures, and it would
be the same if the number was 327,000, or .00327. In general,
count the figures, after cutting off the zeros at either end, unless
they are obtained by the measurement ; thus 300,000 has three sig-
nificant figures if we know that it is more correct than 301,000 or
299,000. In reducing results we should never retain but one more
significant figure than has been obtained in the first measurement,
and must remember that the last of these figures is sometimes
liable to an error of several units.
Successive Approximations. This method is also known as
that of trial and error. It consists in assuming an approximate
value of the magnitude to be constructed, measuring the error,
correcting by this amount as nearly as we can, measuring again,
and so on, until the error is too small • to do any harm. As an
example, suppose we wish to cut a plate of brass so that its weight
shall be precisely 100 grammes. We first cut a piece somewhat
too large, weigh it and measure its area. If its thickness and
density were perfectly uniform we could at once, by the rule of
GRAPHICAL METHOD. 11
three, determine the exact amount to be cut off. As, however, it
will not do to make it too light, we cut off a somewhat less quan-
tity and weigh again ; by a few repetitions of this process we may
reduce the error to a very small amount.
This method is sometimes the only one available, but it should
not be too generally used, as it encourages guessing at results, and
tends to destroy habits of accuracy.
GRAPHICAL METHOD.
Suppose that we have any two quantities, x and y, so connected
that a change in one alters the other. Then we may construct a
curve, in which abscissas represent various values of £c, and ordi-
nates the corresponding values of y. Thus suppose we know that
y is always equal to twice x. Take a piece of paper divided into
squares by equidistant vertical and horizontal lines. Select one of
each of these lines to start from. The vertical one is called the
axis of Y, the other the axis of X, and their intersection, the ori-
gin. Make x = 1, y will equal 2, since it is double x\ now con-
struct a point distant 1 space from the origin horizontally, and 2
vertically. Make x = 2, y = 4, and we have a second point;
x — — 1, gives y = — 2, &c., and x — 0, gives y = 0. Connecting
these points we get a straight line passing through the origin, as is
evident by analytical geometry from its equation, y = 2x. Again,
let y always equal the square of cc, and we have the corresponding
values x = Q, y = 0 ; «j = 1, y = 1 ; x = — 1, y = 1 ; x = 2,
y = 4 ; connecting all the points thus found we obtain a par-
abola with its apex at the origin, and tangent to the axis of JX7
As another example, suppose we have made a series of experiments
on the volumes of a given amount of air corresponding to different
pressures. Construct points making horizontal distances volumes,
and vertical distances pressures. It will be found that a smooth
curve drawn through these points approaches closely to an equi-
lateral hyperbola with the two axes as asymptotes. Now this
curve has the equation xy = a, or y = a -f- ic, that is, the volume
is inversely proportional to the pressure, which is Mariotte's law.
Owing to the accidental errors the points will not all lie on the
curve, but some will be above it and others below, and this will be
true however many points may be observed.
12 INTERPOLATION.
In general, then, after observing any two quantities, A and B,
construct points such that their ordinates and abscissas shall be
these quantities respectively. Draw a smooth curve as nearly as
possible through them, and then see if it coincides with any com-
mon curve, or if its form can be defined in any simple way. To
acquire practice in using the Graphical Method it is well to con-
struct a number of curves representing familiar phenomena, as
the variation in the U. S. debt during the late war, the strength
of horses moving at different rates, and the alterations of the ther-
mometer during the day or year. It is by no means necessary that
the same scale should be used for vertical, as for horizontal dis-
tances, but this should depend on the size of paper, making the
curve as large as possible. The greatest accuracy is attained when
the latter is about equally inclined to both axes.
It is sometimes better when one of the variables is an angle to
use polar coordinates. In this case paper must be used with a
graduated circle printed on it. The points are constructed by
drawing lines from the centre in the direction represented by one
variable, and measuring off on them distances equal to the other.
For ordinary purposes circles may easily be drawn, and divided
with sufficient accuracy by hand. Laying off the radius on the
circumference divides it to 60° ; bisecting these spaces gives 30°,
and a second bisection 15°. By trial these angles may be divided
into three equal parts, which is generally small enough, as the ob-
servations are usually taken at intervals of 5°.
Interpolation. All kinds of interpolation are very readily per-
formed by the Graphical Method. After constructing one curve
to find the value of y, for any given value of x as a/, we have only
to draw a line parallel to the axis of YJ at a distance xf, and note
the ordinate of the point where it meets the curve. Inverse inter-
polation is performed in the same manner, and this method is
equally applicable, whether the observations are at equal intervals
or not. As by drawing a smooth curve the accidental errors are
in a great measure corrected, this method of interpolation is often
more accurate than that by differences.
Residual Curves. The principal objection to the Graphical
Method, as ordinarily used, is its inaccuracy, as by it we can rarely
obtain more than three significant figures, although Regnault, by
RESIDUAL CURVES. 13
using a large plate of copper and a dividing engine to construct
his points, attained a higher degree of precision.
It will be found, however, that in many of the most carefully
conducted researches the fourth figure is doubtful, as for example?
in Regnault's measurements of the pressure of steam, and even in
Angstrom's and Van der Willingen's determinations of wave-
lengths.
By the following device the accuracy of the Graphical Method
may be increased almost indefinitely. After constructing our
points, assume some simple curve passing nearly through them.
From its equation compute the value of y for each observed value
of a, and construct points whose ordinates shall equal the differ-
ence between the point and curve on an enlarged scale, while the
abscissas are unchanged. Thus let a/, yr be the observed coordi-
nates, and y =f(x), the assumed curve. Construct a new point,
whose coordinates are x' and a \_yr — /(«')]> m which a equals 5,
10, or 100, according to the enlargement desired.
Do the same for all the other points, and a curve drawn through
them is called a residual curve. In this way the accidental errors
are greatly enlarged, and any peculiarities in the form of the curve
rendered much more marked. If the points still fall pretty regu-
larly, we may construct a second residual curve, and thus keep on
until the accidental errors have attained such a size that they may
be easily observed. To find the value of y corresponding to any
given value of a?, as a?n, we add/(a;n) to the ordinate of the cor-
responding point of the residual: curve, first reducing them to the
same scale. Most of the singular points of a curve are very
readily found by the aid of a residual curve. See an article by the
author, Journal of the Franklin Institute, LXI, 272.
Maxima and Minima. To find the highest point of a curve,
use, as an approximation, a straight line parallel to the axis of J£J
and nearly tangent to the curve. Construct a residual curve,
which will show in a marked manner the position of the required
point. The same plan is applicable to any other maximum or
minimum.
Points of Inflexion. Draw a line approximately tangent to
the curve at the required point. In the residual curve the change
of curvature becomes very marked.
14 ASYMPTOTES.
Asymptotes. Asymptotes present especial difficulties to the
Graphical Method, as ordinarily used. Suppose our curve asymp-
totic to the axis of -XT; construct a new curve with ordinates
unchanged, and abscissas the reciprocals of those previously used,
that is equal to 1 -j- x. It will contain between 0 and 1 all the
points in the original curve between 1 and QO . It will always pass
through the origin, and unless tangent to the axis of X at this
point the area included between the curve and its asymptote will
be infinite. When this space is finite, it may be measured by con-
structing another curve with abscissas as before equal to 1 -f- x,
and ordinates equal to the area included between the curve and
axis, as far as the point under consideration. Find where this
curve meets the axis of Y, and its ordinate gives the required
area. A problem in Diffraction is solved by this device in the
Journal of the Franklin Institute, LIX, 264.
Curves of Error. This very fruitful application of the Graphi-
cal Method is best explained by an example. Suppose we wish to
draw a tangent to the curve B'A, Fig. 2, at the point A. Describe
a circle with A as a centre, through which
pass a series of lines, as AB, AD, AE.
Now construct C by laying off BC equal
to AB', the part of the curve cut off by
the line. We thus get a curve CD,
called the curve of error, intersecting the
circle at D, and the line AD is the re-
quired tangent. This is evident, since if
we made our construction at this point we
should have no distance intercepted, or the line AD touching, but
not cutting, the curve. A similar method may be applied to a
great variety of problems, such as drawing a tangent parallel to a
given line, or through a point outside the curve.
Three Variables. The Graphical Method may also be applied
where we have three connected variables. If we construct points
whose coordinates in space equal these three variables, a surface is
generated whose properties show the laws by which they are con-
nected. To represent this surface the device known as contour
lines may be used, as in showing the irregularities of the ground
in a map. First, generate a surface by constructing points in which
CONTOUR LINES. 15
ordinates and abscissas shall correspond to two of the variables, and
mark near each in small letters the magnitude of the third varia-
ble, which represents its distance from the plane of the paper. If
now we pass a series of equidistant planes parallel to the paper,
their intersections with the surface will give the required contour
lines. To find these intersections, connect each pair of adjacent
points by a straight line, and mark on it its intersections with
the intervening parallel planes. Thus if two adjacent points have
elevations of 28 and 32, we may regard the point of the surface
midway between them, as at the height 30, or as lying on the 30
contour line. Construct in this way a number of points at the
same height, and draw a smooth curve approximately through
them ; do the same for other heights, and we thus obtain as many
contours as we please.
They give an excellent idea of the general form of the surface,
and by descriptive geometry it is easy to construct sections passing
through the surface in any direction. An easy way to understand
the contours on a map is to imagine the country flooded with wa-
ter, when the contours will represent the shore lines when the
water stands at different heights. This method is constantly used
in Meteorology to show what points have equal temperature, pres-
sure, magnetic variation, &c. Contour lines follow certain general
laws which are best explained by regarding them as shore lines, as
described above. Thus contour lines have no terminating points ;
they must either be ovals, or extend to infinity. Two contours
never touch unless the surface becomes vertical, nor cross, unless it
overhangs. A single contour line cannot lie between two others,
both greater or both smaller, unless we have a ridge or gulley per-
fectly horizontal, and at precisely the height of the contour. In
general, such lines should be drawn either as a series of long ovals,
or as double throughout. There will be no angles in the contour
lines unless there are sharp edges in the original surface. A con-
tour line cannot cross itself, forming a loop, unless the highest
point between two valleys, or the lowest point between two hills,
is exactly at the height of the contour.
The value of contour lines in showing the relation between any
three connected variables, is well illustrated in a paper by Prof. J.
16 PHYSICAL MEASUREMENTS.
Thomson, Proc. of the Royal Society, Nov., 1871, also in Nature,
Y, 106.
To acquire facility in using the Graphical Method, it is well to
apply it to some numerical examples. Thus take the equation
y = axs + &c2 — ex -\- d, assume certain values of a, 5, c and d,
and compute the value of y for various values of x. We thus get
a curve with two maxima or minima, and a point of inflexion.
Find their position first by residual curves, and then by the calcu-
lus, and see if they agree. In the same way the curve ya?2 — Zayx
-}-a2y = #, has the axis of X for an asymptote. Assume, as before,
positive values of a and 6, and determine the area between the
curve and asymptote, first by construction and then analytically.
PHYSICAL MEASUREMENTS.
The measurement of all physical constants may be divided into
the determination of time, of weight and of distance, the appara-
tus used varying with the magnitude of the quantity to be meas-
ured and the degree of accuracy required.
Measurement of Time. A good clock with a second hand, and
beating seconds, should be placed in the laboratory, where it can
be used in all experiments in which the time is to be recorded.
Watches with second-hands do not answer as well, as they gener-
ally give five ticks in two seconds, or some other ratio which ren-
ders a determination of the exact time difficult. The true time
may be measured by a sextant or transit, as described in Experi-
ment 16. This should be done, if possible, every clear day by
different students, and a curve constructed, in which abscissas
represent days, and ordinates errors of the clock, or its deviations
from true time. Short intervals of time may be roughly measured
by a pendulum, made by tying a stone to a string, or better, by a
tape-measure drawn out to a fixed mark. We can thus measure
such intervals as the time of flight of a rocket or bomb-shell, the
distance of a cannon or of lightning, by the time required by
sound to traverse the intervening space, or the velocity of waves,
by the time they occupy in passing over a known distance. After
the experiment we reduce the vibrations to seconds by swinging
our pendulum, and counting the number of oscillations per minute.
MEASUREMENT OF TIME. 17
By graduating the tape properly, we may readily construct a very
serviceable metronome.
Where the greatest accuracy is required, as in astronomical ob-
servations, a chronograph is used. A cylinder covered with paper
is made to revolve with perfect uniformity once in a minute. A
pen passes against this, and receives a motion in the direction of
the axis of the cylinder, of about a tenth of an inch a minute,
causing it to draw a long helical line. An electro-magnet also acts
on the pen, so that when the circuit is made and broken, the latter
is drawn sideways, making a jog in the line. To use this appara-
tus a battery is connected with the electro-magnet, and the pendu-
lum of the observatory clock included in the circuit, so that every
second, or more commonly every alternate second, the circuit is
made for an instant and then broken. Wires are carried to the
observer, who may be in any part of the building, or even at a
distance of many miles, and whenever he wishes to mark the time
of any event, as the transit of a star, he has merely, by a finger
key (such as is used in a telegraph office), to close the circuit,
when it is instantly recorded on the cylinder. When the observa-
tions are completed the paper is unrolled from the cylinder, and is
found to be traversed by a series of parallel straight lines, Fig. 3,
one corresponding to each minute, with indentations corresponding
to every two seconds. The time
may be taken directly from it, the 4 —
fractions of a second being meas-
ured by a graduated scale. One
great difficulty in making this ap-
paratus was to render the motion
of the cylinder perfectly uniform, as if driven by clock-work it
would go with a jerk each second. This is avoided by a device known
as Bond's spring governor, in which a spring alternately retards
and accelerates a revolving axle when it moves faster or slower
than the desired rate. The seconds marks form a very delicate
test for the regularity of this motion, since in consecutive minutes
they should lie precisely in line, and the least variation is very
marked in the finished sheet. It is a very simple matter by this
apparatus to measure the difference in longitude of two points. It is
merely necessary that an observer should be placed at each station,
18 MEASUREMENT OF TIME.
with a transit and finger key, a telegraph connecting them with
the chronograph. They watch the same star as it approaches their
meridian, and each taps on his finger key the instant it crosses the
vertical line of his transit. Two marks are thus made on the chro-
nograph, and the interval between them gives the difference in
longitude. The advantage of this method of taking transits is not
so much its accuracy, as the ease and rapidity with which it is used.
Observers can work much longer with it without fatigue, and can
use many more transit wires, thus greatly increasing the number
of their observations. It is called the American or telegraphic
method, in distinction from the old, or " eye and ear " method of
observing transits, where the fractions of a second were estimated,
as described in Experiment 15.
The chronograph is exceedingly convenient in all physical in-
vestigations where time is to be measured, and nothing but its
expense prevents its more general application.
A simple means of measuring small intervals of time with accu-
racy, is to allow a fine stream, of mercury to flow from a small
orifice, and collect and weigh the amount passed during the time
to be measured. Comparing this with the flow per minute we
obtain the time. A less accurate, but much more convenient,
liquid for this purpose is water, using, in fact, a kind of clepsydra.
Where very minute intervals of time are to be measured they
are commonly compared with the vibrations of a tuning-fork in-
stead of a pendulum. A fine brass point is attached to the fork
which is kept vibrating by an electro-magnet. If a plate of glass
or piece of paper covered with lampblack, is drawn rapidly past
the brass point, a sinuous line is drawn, the sinuosities denoting
equal intervals of time, whose magnitude is readily determined
when we know the pitch of the fork. A second brass point is
placed by the side of the fork and depressed from the beginning to
the end of the time to be measured. The length of the line thus
drawn, compared with the sinuosities, gives the time with great
accuracy. Recently a clock has been constructed, in which the
pendulum is replaced by a reed vibrating one thousand times a
second. The clock is started and stopped, so that it is going only
during the time to be measured, and the hands record the number
MEASUREMENT OF WEIGHT. 19
of vibrations made. The reed produces a musical note, and any
irregularity is at once detected by a change in its pitch.
Measurement of Weight. This is done almost exclusively by
the ordinary balance, whose principle is so fully explained in any
good text-book of Physics that a detailed description is unneces-
sary here. We test the equality in length of its arms by double
weighing, that is, placing any heavy body first in one pan and then
in the other, and seeing if the same weights are required to coun-
terpoise it in each case. The center of gravity should be very
slightly below the knife-edges. If too low the sensibility is dimin-
ished, if too high the balance will overturn, and if coincident with
them the beam, if inclined, will not return to a horizontal position
The three knife-edges must be in line, otherwise the centre ot
gravity will vary with the weight in the scale pans, and of course
the friction must be reduced to a minimum. A high degree of
accuracy may be obtained with even an ordinary balance by first
counterpoising the body to be weighed, then removing it and not-
ing what weights are necessary to bring the beam again to a hori-
zontal position. A spring balance is sometimes convenient for
rough work, from the rapidity with which it can be used. It may
be rendered quite accurate, though wanting in delicacy, by noting
the weight required to bring its index to a certain point, first when
the body to be weighed is on the scale pan, and then when it is
removed.
Measurement of length. Distances are most commonly meas-
ured by a scale of equal parts, that is, one with divisions at regular
intervals, as millimetres, tenths of an inch, &c. This scale is then
placed opposite the distance to be measured, and the reading taken
directly. To obtain greater accuracy than within a single division,
we may divide them into tenths by the eye, as in Experiment 1.
The steel scales of Brown & Sharpe are good for common measure-
ments, and may be obtained with either English or French gradua-
tion. Instead of dividing into tenths by the eye, a vernier is
frequently used. Thus to read a millimetre scale to tenths, nine
spaces are divided into ten equal parts, each of which will be a
tenth of a millimetre less than the divisions of the scale, as in Ex
periment 2.
One of the best devices for measuring very minute quantities is
20 MEASUREMENT OF LENGTH.
the micrometer screw. A divided circle is attached to the head of
a carefully made screw, so that a large motion of the former cor-
responds to a very minute motion of the latter. Thus if the pitch
of the screw is one millimetre, and the circle is divided into one
hundred parts, turning it completely around will move the screw
but one millimetre, or turning it through one division only one
hundredth of a millimetre. One of the best examples of this in-
strument is the dividing engine, which consists of a long and very
perfect micrometer screw with a movable nut. See Experiment 21,
also Jamirfs Physics, I, 25. It is much used in engraving scales,
but it has certain defects which are unavoidable, and have caused
some of our best mechanicians to give it up. For example, it is
impossible to make a screw perfectly accurate, and every joint, of
which there are several, is a source of constantly varying error.
For these reasons, and owing to its expense, the instrument de-
scribed in Experiment 22 is for many purposes preferable. Two
blocks of wood are drawn forward alternately step by step, through
distances regulated by the play of a peg between a plate of brass
and the end of a screw. As all joints are thus avoided, and the
interval is determined by the direct contact of two pieces of metal,
great accuracy is attainable by it.
Where several scales are to be made with the utmost accuracy,
one should first be divided as correctly as possible, and its errors
carefully studied by comparing the different parts with one an-
other, or with a standard. It may then be copied by laying it on
the same support with one of the other scales, and moving both so
that one shall pass under a reading microscope, the other under a
graver. We may thus copy any scale with great accuracy, but the
process is very laborious. A good way to construct the first scale
is by continual bisection with beam compasses, as is done in grad-
uatino- circles. The finest scales are ruled with a diamond on
O
glass. M. Nobert has succeeded in making them with divisions
of less than a hundred thousandth of an inch. The intervals
are so minute that until within a few years no microscope could
separate the lines. The method of making them is kept a secret.
Mr. Peters, by a combination of levers, has succeeded in reducing
writings or drawings to less than one six thousandth their original
size. He exhibited some writing done by this machine, which
MINUTE MEASUREMENTS. 21
was so minute that the whole Bible might be written twenty-seven
times in a square inch. Finally, it is claimed that Mr. Whitworth
was able to detect differences of one millionth of an inch with a
micrometer screw he has constructed.
To measure very minute distances a microscope is often used
with a scale inserted in its eyepiece, which is used like a common
rule. The absolute size of the divisions must be determined be-
forehand by measuring with it a standard millimetre, or hundredth
of an inch. A more accurate method, however, is the spider-line
micrometer, in which a fine thread is moved across the field of view
by a micrometer screw, and small distances thus measured with
the greatest precision. By using two of these instruments, which
are then called reading microscopes, larger distances may be meas-
ured, or standards of length compared, as in Experiment 20.
Small distances are also sometimes measured by a lever, with
one arm much longer than the other, so that a slight motion of the
latter is shown on a greatly magnified scale. Instead of a long
arm it is better to use a mirror, and view in it the image of a scale
by a telescope. An exceedingly small deviation is thus readily
perceptible, and this arrangement, sometimes known as Saxton's
pyrometer, has been applied to a great variety of uses. Where we
wish to bring the lever always into the same position a level may
be substituted for the mirror, forming the instrument called the
contact level. Small distances are also sometimes measured by a
wedge with very slight taper, but this plan is objectionable on
many accounts. In geodesy all the measurements are dependent
on the accurate determination in the first place of a distance of
five or ten miles, called a base line. Most of the above devices
have been tried on such lines ; thus the reading microscope was
used by Colby in the Irish survey, the wedge in Hanover, and by
Bessel in Prussia, the lever by Struve in Russia, and the contact
level is now in use on our Coast Survey. The principle in all is to
use two long bars alternately, which are either brought in contact,
or the distance between their ends measured each time they are
laid down.
Many other physical constants are really determined by a meas-
ure of length. Thus temperatures are determined by a scale of
equal parts in the thermometer, and here sufficient accuracy is ob-
22 AREAS AND VOLUMES.
tained by reading with the unaided eye. Pressures of air and
water are also measured by the height of a column of mercury or
water. Where great accuracy is required, as in the barometer, a
vernier is commonly used.
The instrument known as the cathetometer is so much used for
measuring heights that it needs a notice here. It consists of a
small telescope, capable of sliding up and down a vertical rod
to which a scale is attached. The difference in height of any two
objects is readily obtained by bringing the telescope first on a level
with one, and then with the other, and taking the difference in the
readings. A level should be attached to the telescope to keep it
always horizontal, but the great objection to the instrument is that
a very slight deviation in its position, which may be caused by
focussing or turning it, is greatly magnified *in a distant object. A
good substitute for this instrument may be made by attaching a
common telescope to a vertical brass tube, the scale being placed
near the object to be measured instead of on the tube, as in Ex-
periment 12.
Although the measurement of the following quantities is directly
dependent on the above, yet their importance justifies a separate
notice.
Measurement of Areas. It is difficult in general to determine
an area with accuracy, especially where it forms the boundary of a
curved surface. If plane, any of the methods of mensuration used
in surveying may be adopted. Of these the best are division into
triangles, Simpson's rule, and drawing the figure on rectangular
paper and counting the number of enclosed squares, allowing for
the fractions. Another method sometimes useful is to cut the figure
out of sheet lead, tin foil, or even card board, and compare its
weight with that of a square decimetre of the same material.
Measurement of Volumes. These are generally determined by
the weight of an equal bulk of water or mercury, using the latter
if the space is small. The interior capacity of a vessel is meas-
ured by weighing it first when empty, and then when filled with
the liquid, as in Experiment 19. The difference in grammes
gives the volume in cubic centimetres when water is used, but
with mercury we must divide by 13.6, its specific gravity. In the
same way we may determine the exterior volume of any body by
ANGLES. 23
immersing it and measuring its loss of weight, as when determin-
ing its specific gravity.
An easier, but less accurate, method is by a graduated vessel.
These are made by adding equal weights or volumes of liquid,
successively, and marking the height to which it rises after each
addition. The volume of any space may then be found by filling
it with water, emptying it into the graduated vessel and reading
the scale attached to the side of the latter.
Measurement of Angles. Angles are measured by a circle di-
vided into equal parts, the small divisions being determined by
verniers or reading microscopes, as in measuring lengths.' A
great difficulty arises from the centre of the graduation not coin-
ciding with that of the circle, and on this account it is best to
have two or more at equal intervals around the circumference. By
taking their mean we eliminate the eccentricity.
The precision of modern astronomy is almost entirely due to
the methods of determining angles with accuracy. This is de-
pendent on two things; first, a good graduated circle, and sec-
ondly, a means of pointing a telescope in a given direction, as
towards a star, with great exactness. The latter is accomplished
by placing cross-hairs at the common focus of the object glass
and eye-piece, so that they may be distinctly seen in the centre of
the field at the same time -as the object. Most commonly two
cross-hairs are used at right angles, one being horizontal, the other
vertical. When, however, we are to bring them to coincide with
a straight line, as in the spectroscope, or in a reading microscope,
they are sometimes inclined at an angle of about 60°, that is, each
making an angle of 30° with the line to be observed. The latter
is then brought to the point of the V formed by their intersection.
Still another method is to use two parallel lines very near together,
the line to be observed being brought midway between them.
The lines may be made of the thread of a spider, of filaments of
silk, of platinum wire, or better for most purposes, by ruling fine
lines on a plate of thin glass with a diamond, and inserting it at
the focus.
There are two methods of graduating circles with accuracy.
The first, which is used in Germany, consists in a direct compari-
son with an accurately divided circle, as when copying scales as
24 GRADUATING CIRCLES.
described above. That is, both circles are mounted on the same
axis, and the divisions of the first being successively brought un-
der the cross-hairs of a microscope, the graver cuts lines on the
second at precisely the same angular intervals. In the second
method, which is much quicker but less accurate, the circle is laid
on a toothed wheel which is turned through equal intervals by
a tangent screw. Both methods are really only means of copy-
ing an originally divided circle, as it is called, and the con-
struction of this with accuracy is a matter of extreme difficulty.
It is dependent on the following principles. Any arc or distance
may be accurately bisected by beam compasses ; the chord of 60°
equals the radius, and the angle 85° 20', whose chord is 1.3554, by
ten bisections is reduced to 5'. By constructing an accurate scale,
laying off 1.3554 times the radius on the circumference, and
repeatedly bisecting the arc, we finally divide the circle into
5' divisions. Where great accuracy is not required we may
divide circles approximately by hand, as described under the
Graphical Method, or more accurately by a table of chords and a
pair of beam compasses. When the divisions of the circle are
very large we may subdivide them by
123 a scale instead of a vernier. Thus if
AJ 1 — |NI|M[ I LB ^g,Fig.4, is part of a circle divided
J? 30 J, into degrees, we may attach a scale
Fig 4m CD, divided to ten minutes, and sub-
divide these into single minutes by
the eye. Thus in Fig. 4 the reading is 2° 35'. Much labor is thus
saved where the circles have to be divided by hand.
Saxton's pyrometer, described above, is of the utmost value in
measuring small angular changes. As the reflected beam moves
twice as fast as the mirror, the accuracy is doubled on this account.
If the scale is flat, allowance must be made for the greater distance
of its ends than the centre. To reduce the reading to degrees and
minutes, the formula, tan 2a = s -f- d is used, or a = .5 tan~ *
s -r- df, in which a is the angle through which the mirror turns, s
the reading, and d the distance of the scale taken in the same
units. Instead of a telescope a light shining through a narrow slit
is sometimes used, and an image projected on the scale by a lens,
or the mirror itself may be made concave. This plan is adopted
RADIUS OF CURVATURE. 25
in the Thomson's Galvanometer, and other instruments for meas-
uring the deviations of the magnetic needle.
Very small angles may also be measured by a spider line mi-
crometer attached to the eye-piece of a telescope. This is used to
determine the distance apart of the double stars, and other minute
astronomical magnitudes. There are other methods, such as di-
vided lenses, double image prisms, &c., but they will be considered
in connection with the particular experiments which serve to illus-
trate them.
Measurement of Curvature. To measure the radius of a sphere,
as the surface of a lens, an instrument called the spherometer is
used. It consists of a micrometer screw at the centre of a tripod,
whose three legs and central point are brought in contact with the
surface. By noting the position of the screw, the radius is readily
computed, as in Experiment 14.
When the surface is of glass, and the curvature very slight, a
much more delicate method is as follows : Focus a telescope on a
distant object, and then view the image reflected in the surface to
be tested. If the latter is concave, it will render the ray less di-
vergent, and hence the eye-piece will have to be pushed in. The
opposite effect is produced by a convex mirror. The amount of
change affords a rough measure of the curvature. This method is
so delicate as to show a curvature whose radius is several miles.
GENERAL EXPERIMENTS.
1. ESTIMATION OF TENTHS.
Apparatus. Two scales, N and j&f, are placed side by side, one
being divided into millimetres, the other into tenths of an inch.
Also a steel rule A, Fig. 5, divided into millimetres, and so ar-
ranged that it may be pushed past a fixed index .Z?, by a microm-
eter screw, C. A spring, D, is used to bring it back, when the
screw is turned the other way.
Experiment. Read the position of each tenth of an inch mark
of scale M, in tenths of a millimetre, estimating the fractions by
the eye. Thus if the interval is one half, call it .5, if a little less,
.4, if not quite a third, .3, and so on for the other fractions. The
.3 and .7 are the hardest to estimate correctly, as we are liable to
imagine the former too great, the latter too small. They should
always be compared with the fractions one and two thirds. Re-
cord your observations in five columns, placing in the first the
readings of the scale M,
in the second the corre-
sponding readings of N,
and in the third the first
differences of N. Next,
subtract the first from the
last number in column two,
J
and divide the difference
by the number of spaces
measured, that is, the num-
ber of readings minus one.
This gives the average dif-
ference, and should be equal to each number of golurnn three.
Subtract it from these numbers, and place the results or errors,
with proper signs, in column four. Next, compute the probable
Fig. 5.
28 ESTIMATION OF TENTHS.
error (see page 3) of a single observation, using the fifth column
for the squares of column four. In this way you can read any
scale much more accurately than by its single divisions, and your
computed probable error shows how closely you may rely on the
result.
Next bring one of the millimetre marks of A, Fig. 5, opposite
the index B. Read its position, as described on page 20. The
scale E gives units, or number of revolutions, and the divided
circle hundredths. Move the screw, set again, and repeat several
times. Take the mean and compute the probable error of a single
observation. Do the same with the next millimetre mark. Now
move the scale until the reading shall be in turn .1, .2, .3, &c.,
of a millimetre, taking care to move the screw after each, so that
you will not be biassed by your previous reading. Next compute
what should be the true readings in these various positions. Thus
let m' be the mean for the first millimetre, m" for the second ; the
reading for one tenth would be mf + (m" — m') -f- 10, for two
tenths m' + 2(m" — m') -r- 10, and so on. See how these read-
ings agree with those previously found. If any differ by a consid-
erable amount repeat them until you can estimate any fraction
with accuracy. This work must be carefully distinguished from
guessing, since there should be no element of chance in it, but an
accurate division of the spaces by the eye. By practice one can
read these fractions almost as accurately as by a vernier.
2. VERNIERS.
Apparatus. A number of verniers and scales along which they
slide are made of large size. The best material is metal or wood,-
although cardboard will do. By making them on a large scale, as
a foot or more in length, there is no trouble in attaining sufficient
accuracy. Several different forms are given in Gillespie's Land
Surveying, p. 228, from which the following may be selected.
1st, Fig. 225, Scale divided to .1, Vernier reads to .01 ; 2d, Fig.
227, Same Vernier retrograde ; 3d, Fig. 228, Scale .05 ; Vernier
.002; 4th, Fig. 229, Scale 1°, Vernier 5'; 5th, Fig. 230, Scale 30',
Vernier 1'; .6th, Fig. 233, scale 20', Vernier 30"; 7th, Fig. 239,'
Scale 30', Vernier 1' ; Double Compass Vernier.
Experiment. A vernier may be regarded as a simple enlarge-
ment of one division of the scale. Thus if the scale is divided
INSERTION OF CROSS-HAIRS. 29
into tenths of an inch, and the vernier into ten parts, it will read
to hundredths of an inch. Always read approximately by the
zero of the vernier, taking the division of the scale next below it.
The fraction to be added is found by seeing what line of the vernier
coincides most nearly with some line of the scale. Thus in the
first example, we obtain inches and tenths by seeing what division
of the scale falls next below the zero of the vernier. If this is 8.6,
and the division marked 7 of the vernier coincides with a line of
the scale, the true reading is 8.6 + .07 = 8.67. To prove this, set
the zero of the vernier at 8.6 exactly. Nine divisions of the scale
equal ten of the vernier. Hence each division of the latter equals
.09, or is shorter by .01 than one division of the scale. Accord-
ingly the line marked 1 of the vernier falls short by .01 of the
scale-division, the 2 line .02, and so on. If we move the vernier
forward by these amounts these lines will coincide in turn. Hence
when the 7 line coincides, as in the above example, it denotes that
the vernier has been pushed forward .07 beyond the 8.6 mark.
This method may be applied to reading any vernier. To find the
magnitude of the divisions of the latter, divide one division of the
scale by the number of parts contained in the vernier.
Read and record the verniers as now set. Then set them as
follows: 1st, 8.03; 2d, 29.9; 3d, 30.866; 4th, 4° 10' 5 5th, 0° 17';
6th, 2°58/30"; 7th, 2° 51'.
The last vernier is a double one, reading either way, the left
hand upper figures being the continuation of those on the lower
right hand. This is best understood by moving it 5' at a time and
noting what lines coincide.
After each exercise the instructor should set all the verniers, and
compare the record of the student with his own.
3. INSERTION OP CROSS-HAIRS.
Apparatus. Some common sewing silk, card-board and muci-
lage, also a pair of dividers, ruler and triangle.
Experiment. A great portion of the accuracy attained in mod-
ern astronomical work is dependent on the exactness with which
we can point a telescope, or other similar instrument, in a given
direction. This is accomplished by inserting two filaments of silk
30
INSERTION OF CROSS-HAIRS.
or spider's web at right angles to each other, at the point within
the telescope where the image of the object is formed. In the
astronomical telescope, where a positive eye-piece is used, this
point lies just beyond the eye-piece, that is between it and the
object-glass. A ring is placed at this point on which the lines are
stretched. In telescopes rendering objects upright, as in most
surveyor's transits, the lines are commonly placed between the
object-glass and erecting lenses, and close to the latter. In the
microscope, and other instruments where a negative eye-piece only
is used, the lines have to be placed on the diaphragm between the
field- and eye-lenses. This plan is objectionable, since the lines
should be very accurately focussed, which can then only be done
by screwing the eye-lens in or out. In the other cases the whole
eye-piece maybe slid in or out until the lines are perfectly distinct,
and do not appear to move over the object when the eye is moved
from side to side.
It is comparatively easy to insert the lines on their ring, where
a positive eye-piece is used. The following experiment therefore
includes the others. Take a negative eye-piece,
Fig. 6, from a microscope or telescope, and unscrew
the eye-lens A. C is the diaphragm which limits
the field of view, and on which the lines should be
placed. Cut from the cardboard a ring, Fig. 7,
whose inner diameter is a little greater than the
opening of the diaphragm, and the outer diameter
such that it will easily rest on C. Mark on it two
Fig. 6.
lines at right angles to each other passing through its centre. Un-
ravel a short piece of the silk thread until you have separated a
single filament. This is best done by holding
the thread with the forceps over a sheet of white
paper. We now wish to stretch two of these
filaments over the lines marked on the card-
board circle. Put a little mucilage on the lat-
ter, dip one end of the silk into it, and press it
down with one of the radial strips of paper
shown in Fig. 7. When this is nearly dry fasten the other end in
the same way, taking care to stretch it so that it shall be straight,
or the twist in the thread will give it a sinuous form. Attach the
Fig. 7.
SUSPENSION BY SILK FIBRES. 31
other thread in the same way, and bending the four strips of paper
down lay the cardboard on the diaphragm. To hold it in place
cut a strip of cardboard or brass, and bending it into a circle push
it into the tube. By its elasticity it will hold the paper strips
firmly against the sides of the tube. If the experiment has been
well performed, on replacing the eye-lens we see two straight lines
at right angles, dividing the field of view into four equal parts.
The cardboard should not project beyond the diaphragm, or it will
give a rough edge to the field of view, and we must be careful that
no mucilage adheres to the visible portions of the threads.
4. SUSPENSION BY SILK FIBRES.
Apparatus. The best method of suspending a light object so
that it shall move very freely is by a single filament of silk. The
only apparatus needed is a stand seven or eight inches high, some
unspun silk (common silk thread will do, but is not so good) and
some fine copper wire. We also need two pairs of forceps, such as
come with cheap microscopes, some bees-wax and a sheet of white
paper.
Experiment. Lay the silk on the paper and pick out a single
fibre a little over six inches long. Bend pieces of the wire into
the shapes A and J?, Fig. 8. Pass one end of the
filament through the ring of J?, and fasten it with . P %
a little wax, twisting or tying it to prevent slip- " u
ping. Fasten the other end to A in the same way, Fi g
making the distance from A to B just six inches.
Hook A into the stand, and lay the object to be suspended, as a
needle on IB.
5. TEMPERATURE CURVE.
Apparatus. A beaker, stand and burner, by which water can
be heated, a Centigrade thermometer, and a clock or watch giving
seconds.
Experiment. Place the thermometer in the water and record
the temperature, dividing the degrees to tenths, as described in
Experiment 1. Place the burner under the beaker at the begin-
ning of a minute, and at the end record the temperature ; repeat
at the end of each minute, as the water is warmed, until the ther-
32
TESTING THERMOMETERS.
mometer stands at 95° ; at the end of the next minute remove
the burner and the temperature will at first continue to rise, and
will then fall rapidly. Record the time (in minutes and seconds)
of attaining 95°, 90°, 85°, &c., taking shorter intervals as the tem-
perature becomes lower, and the cooling less rapid. Record your
results in two columns, one giving times, the second temperatures.
Finally construct a curve in which abscissas represent times, and
ordinates temperatures, making in the former case, one space equal
one minute, in the latter, one degree.
When two students, A and _Z?, are engaged in this experiment,
the following system should be used. A observes the watch and
records, while B attends to the thermometer. Five seconds be-
fore the minute begins A says, Heady ! and at the exact begin-
ning, Now! JB then gives the reading which A record's. This
plan saves much trouble, and greatly increases the accuracy of any
observations which must be made at regular intervals of time.
6. TESTING THERMOMETERS.
Apparatus. An accurate Centigrade thermometer is hung upon
a stand, and close to it a Fahrenheit thermometer, which is to be
tested, their bulbs being at the same height, and close together.
A telescope with which they can be read more accurately is placed
on a stand at a short distance, and their temperature may be al-
tered at will by immersing their bulbs in a beaker of water, which
may be either cooled by ice, or heated by a Bunsen burner. Some
arrangement is desirable for stirring the water to keep it at a uni-
form temperature. One way is to use a circular disk of tin with
holes cut in it, which may be raised or lowered in the beaker by a
cord passing over a pulley, so that the observer, while looking
through the telescope, can stir the water by alternately tightening
and loosening the cord. A simple glass stirring rod may be used
instead, if preferred.
Experiment. The problem is to determine the error of the
Fahrenheit thermometer at different temperatures, by comparing
it with the Centigrade thermometer, which is regarded as a stand-
ard. By means of the telescope read them as they hang in the
air, estimating the fractions of a degree in tenths. Do the same
when their bulbs are immersed in water, then cool them with ice
and read again. This observation is important, as it shows the
absolute error of each instrument. Next heat the water a few
ECCENTRICITY OF GRADUATED CIRCLES. 33
degrees with the burner, and then remove the latter. The tem-
perature will still rise for a short time, then become stationary and
fall. Read each thermometer at its highest point, stirring the wa-
ter meanwhile. Repeat at intervals of about 10° until the water
boils, and finally immerse again in the ice water, and see if the
reading is the same as before.
We have now two columns of figures, the first giving the tem-
perature of the Centigrade, the second that of the Fahrenheit,
thermometer. Reduce the first to the second, recollecting that
0° C. = 32° F., and 100° C. = 212° P.; hence F. = f C. + 32°,
calling C and F the corresponding temperatures on the Centi-
grade and Fahrenheit scales respectively. Write the numbers
thus found in a third column, and the errors will equal the differ-
ences between them and the readings given in column two. If
the Centigrade thermometer does not stand at zero when im-
mersed in ice water, all its readings should be corrected by the
amount of the deviation, taking care to retain the proper sign.
Now construct a curve whose ordinates shall represent the errors
on an enlarged scale, and abscissas the temperatures.
7. ECCENTRICITY OF GRADUATED CIRCLES.
Apparatus. A circle divided into degrees carries a pointer
with an index at each end, which turns eccentrically, that is, the
centres of the pointer and circle do not coincide. It may be made
in a variety of ways. One of the simplest is to place a pivot on
one side of the centre of the circle, and on it a rod with a needle
projecting from each end. Another way is to let the circle turn
and cover it with a plate of glass, on which are marked two fine
lines, with a diamond or India ink. The indices may also be made
of fine wire, or horsehair. Lines of consid-
erable length must be used, since the edge
of the circle advances and recedes as it is
turned. If greater accuracy is desired the
plan shown in Fig. 9 may be adopted. The
two. indices (which, may have verniers) are
connected with the centre by the arms A G
and CB. The circle turns around the pin
J9, and a rod passing through the guides
EF, keeps the verniers in the proper posi-
tion. Another good instrument for this experiment is the form of
compass described under Magnetism in the latter part of the pres-
ent work.
34 CONTOUR LINES.
Experiment. Set the index A at 0° by turning the circle, and
read £. Repeat moving A 10° at a time, until a complete revo-
lution has been made. We have now two columns, giving the
corresponding readings of A and JB. Subtract 180° from the lat-
ter, and \(A + B — 180°), or \(A + JB) — 90° will be the true
reading ; write this in column three ; in the same way the error of
each index is \(A — B) — 90°, which should be written in the
fourth column. Construct a curve with abscissas equal to the
numbers in column three, and ordinates equal to those in column
four, enlarged. At the highest and lowest parts of the curve the
indices differ most from their true position, or the absolute error,
if we read one only, is here greatest. Find these points by Curves
of Error, p. 14. On the other hand, where the curve cuts the axis
the two indices are opposite each other, and the abscissa gives the
azimuth of the line CD. As the ordinates alter most rapidly at
these points, the error, when reading a small angle by one index, is
here a maximum. Draw tangents, as before, by Curves of Error,
and from their direction we can compute the amount of variation.
It is a very good exercise to deduce by trigonometry the theoreti-
cal curve, and constructing it on the same sheet of paper to com-
pare the results with those obtained by your measurement.
We have heretofore supposed that the line connecting the in-
dices passed through the axis around which they turned, or that
D lies on EF. If, as often happens in practice, this is not the
case, a second correction is necessary.
8. CONTOFK LINES.
Apparatus. No apparatus is needed for this experiment, except
ordinary writing materials. It is, in fact, an exercise rather than
an experiment.
Experiment. Mark in your note book nine rows of six points
each, so as to form forty squares of about one inch on a side.
Mark them with numbers taken from the adjoining table A. Now
suppose these numbers represent the heights of the points to which
they are attached, and we wish to draw contour lines to show the
form of the surface passing through them. As the points are
pretty near together we may assume that a line connecting any
CLEANING MERCURY.
35
two that are adjacent will lie nearly in the surface. Now regard
your drawing as a map, as on p. 15, and suppose the ground
B
83
79
73
79
79
74
46
56
67
84
86
84
66
76
68
57
40
28
82
78
70
81
84
76
29
52
73
94
'86
73
73
72
50
29
28
52
78
76
66
83
88
73
39
31
65
82
.70
56
66
48
31
11
27
41
74
73
58
78
82
63
60
62
68
72
57
49
59
29
29
42
38
29
70
61
50
73
82
74
69
73
81
81
65
48
38
46
38
72
61
39
71
58
61
82
96
75
80
94
80
81
73
50
27
35
70
99
70
28
70
59
70
83
84
72
80
58
58
65
70
49
21
46
87
96
60
29
67
65
72
79
73
69
67
58
58
67
62
46
33
63
95
81
49
31
66
67
72
76
69
75
74
68
72
80
49
37
44
71
86
64
47
27
flooded with water to a height of 80. Evidently all the points in
the upper line will be submerged except that on the left, and the
shore line will come between 79 and 83, about a fourth way from
the former. Also midway between 82 and 78 in the second line,
two fifths of the way from 78 to 83, and a third way from 79 to
82. Several points are thus obtained in each square through which
the contour line passes. After obtaining as many as possible,
draw a smooth curve nearly coinciding with them all, paying
special attention to the rules given under the Graphical Method.
Construct in the same way other contours at intervals of ten units.
Do the same with the numbers in table B or C.
This work is very well supplemented by procuring from the
U. S. Signal Bureau at Washington, some of their blank maps
(issued at $2.75 per 100), and filling them out from the weather
reports for the day, according to their published directions. These
maps may also be used for drawing isothermals, isogonals, &c., if a
list is prepared in the first place of the temperature, magnetic
variation, &c., of a large number of stations in the United States.
The method adopted for drawing these lines is essentially the same
as that given above, only the points are irregularly spaced.
9. CLEANING MERCURY.
Apparatus. But little apparatus is needed for this experiment,
except such as is found in every chemical laboratory. Some bot-
tles, funnels, &c., should be placed on the table, and the student
should try as many of the following methods of purification as he
can, and record in his note-book his opinion of their comparative
value.
36 CLEANING MERCURY.
Experiment. Mercury is so much used in physical experiments
that every student should know how to clean it. The impurities
may be divided into three classes : first, mixture with metals, es-
pecially lead, zinc and tin ; secondly, common dust and dirt; and
thirdly, water or other liquids.
Redistillation is almost the only way to remove the metals, and
even this is not perfectly effectual, especially in the case of zinc.
Moreover, by long boiling a small amount of oxide is formed,
which is dissolved by the metal. The mercury used for amal-
gamating battery plates should therefore be kept separate from the
rest and used for this purpose only. If but little of the metal is
present it may be removed by agitating with dilute nitric acid.
The best way to do this is to fill a long vertical tube with the acid
and allow the mercury to flow into it from a funnel, in which is a
paper filter with a fine hole in the bottom. The mercury falls
through the long column of liquid in minute globules, and is thus
readily and thoroughly cleaned. It may be drawn out below by
a glass stopcock, or by a bent tube in which a short column of
mercury shall balance a long column of acid. As the mercury
collects it flows out of the end of the tube into a vessel placed to
receive it. Instead of nitric acid a solution of nitrate of mercury
may be used, if preferred. Another method is to fill a bottle
about a quarter full of mercury, add a quantity of finely powdered
loaf sugar, and shake violently. The metallic impurities are ox-
idized at the expense of the air, which must be renewed by a pair
of bellows.
A great variety of devices are used to remove the mechanical
impurities of mercury. For example, pouring it into a bag of
chamois leather and squeezing the latter until the mercury comes
through in fine globules. Or, making a needle hole in the point
of a paper filter, placing it in a funnel and letting the mercury run
through. The mercury may be washed directly with water, by
shaking them together in a bottle, or better, filling a jar half full
of mercury and letting the water from the hydrant bubble up
through it. This is an excellent way to remove most liquids.
Next, to remove the water, pour the mixture into a small bottle,
when the mercury will settle to the bottom, and the water over-
flow from the top. When the mercury fills the bottle transfer it
CALIBRATION BY MERCURY. 37
to another vessel and repeat. If there is only mercury enough to
half fill the bottle the second time, pour back some of the mercury
already dried to displace the remaining water. Another way is to
close the end of a funnel with the finger and pour in the mixture,
drawing off the mercury below and leaving the water above. Care
must be taken that the mercury does not spurt out on one side
and escape. An inverted bottle, or better, a vessel with a tube
and stopcock below, is more convenient for this purpose.
When only a few drops of water are present they may be re-
moved by blotting paper, or a camel's hair brush. Also by apply-
ing heat ; but in this case a stain will be left when the water evap-
orates, unless it has been previously distilled.
To see if the mercury is pure pour it into a porcelain evaporat-
ing dish. If lead is present it will tarnish the sides. A thin film
will also, after a short time, form on its surface, due to oxidation ;
zinc and tin produce a similar effect. The surface when at rest
should be very bright and almost invisible, and small globules, if
detached, should be perfectly spherical, and not adhere to the glass
but roll over it when the surface is inclined.
10. CALIBRATION BY MERCURY.
Apparatus. The best way to perform this experiment is that
given by Bun sen in his Gasometry, p. 27. This method is sub-
stantially as follows : Select a glass tube, about 2 cm. in diameter,
and 40 cm. long, closed at one end. Fasten to it a paper mil-
limetre scale. This is placed upright in a stand, at a short dis-
tance from a small telescope, by which the scale may be read with
accuracy. On another stand is placed a vessel containing about
two kilogrammes of pure mercury, covered with a layer of concen-
trated sulphuric acid, with a stopcock below, by which it may be
drawn off. A small glass tube, also closed at one end, is used to
receive it, which should contain, when filled, about 10 cm.8 Its
open end is ground flat, and it may be closed with a plate of
ground glass, which is fastened to the thumb by a piece of rubber.
Experiment. Both mercury and tube should be perfectly clean,
but if not, a few drops of water may be placed in the longer tube,
provided great accuracy is not required. Fill the small tube with
mercury, holding it with the fingers of the left hand, and remove
the surplus by pressing the glass plate, which should be attached
to the left thumb, down on to it. Take care that no air bubbles
38 CALIBRATION BY MERCURY.
are imprisoned. Empty the mercury into the large tube, and read
its height on the scale by the telescope, measuring from the top
of the curved surface of the liquid. A clean wooden rod may be
used to remove any bubbles of air or globules of mercury which
adhere to the sides of the tube. Repeat this operation until the
large tube is full of mercury. We now wish to know the volume
of the small tube, as this is the unit in terms of which the larger
one has been calibrated. The most accurate way to do this is to
weigh the whole amount of mercury transferred, and divide by
the number of times the smaller tube has been filled. But as it is
generally difficult to weigh so heavy a body accurately, the con-
tents of the smaller tube had better be weighed alone, repeating
two or three times to see how much the quantity used will vary in
consecutive fillings. The volume is then obtained by dividing the
weight by 13.6, the specific gravity of mercury. Multiplying the
quotient by 1, 2, 3, 4, &c., we obtain the volumes corresponding to
our observed readings of the mercury column in the long tube.
Represent the results by a residual curve, as follows : Let s be
the scale reading when the small tube has been emptied once into
the long tube, and / when the latter is full, or has received n times
this volume of mercury, which we will call v. Then (n — l)v of
mercury will fill the space sf — s, and the average volume per unit
of length will equal (n — l)u -r- (/ — s) = a. If the tube was
perfectly cylindrical we could find the volume V for any scale
reading S by the formula, V = a {S — s) -f v. In reality the
tube is probably a little larger in some places than in others,
it is therefore better to retain only two significant figures in a,
and then compute by the formula the volumes corresponding to
the various scale readings that have been observed. Subtract
each of these from the corresponding volumes 1, 2, 3, &c., times v,
and construct a residual curve in which ordinates equal these dif-
ferences on an enlarged scale, and abscissas the scale readings.
We can now obtain the volume with the greatest accuracy for any
scale reading by adding to the value of V given by the formula,
the ordinate of the corresponding point of the curve. A table
may thus be constructed, giving the volume corresponding to each
millimetre mark of the scale. But it is generally sufficiently accu-
rate to make a simple interpolation from the original measurements,
CALIBRATION BY WATER. 39
using only the first differences, as when employing logarithmic
tables.
11. CALIBRATION BY WATER.
Apparatus. A Mohr's burette B, Fig. 10, on a stand, and the
vessel to be graduated A, which should be about six inches high,
and an inch and a half in diameter. A paper scale divided into
tenths of an inch should be attached to A with gum tragacanth,
although shellac, or even mucilage, answers tolerably. A long
string wound spirally around the vessel will keep the scale in place
until the gum is dry.
Experiment. Fill the burette JB to the zero mark. This is
done by adding a little too much water, and
drawing it off by the stopcock C into another
vessel, until it stands at precisely the right level.
Next, let the water flow into A until it reaches
the one tenth of an inch mark, and read J5. Do
the same for each tenth of an inch, until the one
inch mark is reached, and then for every half
inch to the top. Do not let the water level in J3 :DQ
^7
fall below the 100 cm.3 mark, but when it ' —
reaches this point refill as before, and add 100 to Fi£- 10-
the volume measured. Care should be taken not to get too much
water into A ; should this happen, a little may be drawn out with
a pipette and replaced in J?, but a slight error is thus introduced.
We have now a series of volumes corresponding to various scale
readings. Construct a curve with these two quantities as co-
ordinates. Find the point of the curve for which the volume is in
turn 10, 20, 30, &c., cm.8, and record the corresponding scale-
reading. If the vessel is to be used for the measurement of vol-
umes cover it with wax and draw horizontal lines on the latter,
having the scale readings just found. Subject it to the fumes of
fluorhydric acid, formed by mixing powdered fluor spar and con-
centrated sulphuric acid. The lines will thus be permanently
etched on the glass.
12. CATHETOMETER.
Apparatus. A Cathetometer may be made by using as a base
the tripod of a music stand or photographer's head-rest, and screw-
40
CATHETOMETER.
ing into it a tube or solid rod of brass. To this is attached a small
telescope with a clamp and set screw, and some form of slow
motion. The latter may be obtained by placing the telescope on
a hinge and raising and lowering one end by a screw. The slight
deviation from a horizontal position will not affect the results, as
the instrument is here used.
At a distance of five or ten feet is placed a U tube, open at both
ends, with one arm about ten inches long, the other forty. The
bend in the tube is filled with mercury, and water is poured into
the long arm. We then have a long column of water sustaining a
short column of mercury, the heights being inversely as the densi-
ties. By the side of this tube is a barometer, made by closing a
common glass tube at one end, filling with mercury, and inverting
over a cistern containing the same liquid. The precautions and
details will be found under Experiment No. 55. By the side of
this tube is placed a rod about ten inches long, sharply pointed at
both ends, and capable of moving up and down so as to touch the
surface of the mercury in the barometer cistern. A steel scale
divided into millimetres is adjacent to both tubes, so that it can be
read at the same time as thg mercury columns.
Experiment. Focus the telescope so that both scale and mer-
cury are distinctly visible.
Then raise it until it is
nearly on a level with A,
the top of the column of
water, and bring its hori-
zontal cross-hair exactly to
coincide by the slow mo-
tion. Read the scale, di-
viding the millimetres into
tenths by the eye. Do the
same at JB and (7; then the
difference in height of A
I ™* i and (7, divided by that of
C and B, will equal the
specific gravity of the mer-
cury, which should be compared with its true value. As the sur-
face of mercury is curved upwards, that of water downwards, the
cross-hairs should be brought to the top of the former, and to the
bottom of the latter. If great accuracy is required in this experi-
ment, allow for the meniscus, or curved portion at the top of the
Fig. 11.
HOOK GAUGE. 41
water, by adding one half its thickness to the height of the water
column.
Next raise the rod EF, and read the height, first of the top
and then of the bottom. The difference will be its length.
It is safer to test the result by moving it and repeating. Then
bring the rod so that it shall just touch the surface of the mercury,
that is, so that the point and its reflection shall coincide, and read
the height of Z>, and of the top of the rod. Their difference
added to the length of the rod gives the height of the column.
Read the height of the standard barometer placed among the
meteorological instruments. Reduce this to millimetres, and sub-
tract from it the other measurement. The difference will be the
depression caused by air and the other errors in the barometer D.
13. HOOK GAUGE.
Apparatus. A stand, Fig. 12, on which may be placed a vessel
of water A, and a micrometer screw j5, by which we can raise or
lower a rod carrying two points, one turned upwards, the other
downwards.
Experiment. Fill up the vessel until the water just covers the
point of the hook. Then turn the screw so that
upon looking at the reflection on the surface of
some object as a window sash, a slight distortion is
produced by the elevation of the water above the
hook. Make ten measurements, moving the
screw after each, take their mean and compute the
probable error of a single observation. When
the point is raised it draws the liquid with it.
Screw it down until it touches the liquid, and C
read the micrometer, then raise it until the liquid Flg< 12'
separates, and take ten readings in each position. Compute, as be-
fore, the probable error, and reduce to fractions of a millimetre,
which is easily done if the pitch of the screw is known. This
gives a measure of the comparative accuracy of the hook and
simple point. Both are used for determining the exact height of
any liquid surface, the hook being employed most frequently in
this country, the point abroad. When the surface of a liquid is
42 SPHEROMETER.
rising or falling, and we wish to know the exact time when it
reaches a given level, we should use the hook when it descends,
otherwise the point ; because the former should always be brought
up to the surface, the latter down to it.
This instrument is so extremely delicate that it will show the
lowering of a surface of water in a few minutes by evaporation.
A variety of interesting researches may be conducted with it, by
the different students of a class. Thus its comparative accuracy
with water, mercury and other liquids, may be measured, their
rate of evaporation, and the effect of impurities, such as a drop of
oil. The height to which a liquid may be raised by the point, is
also a test of its viscosity.
14. SPHEROMETER.
Apparatus. Two lenses, one convex, the other concave, a piece
of thick plate glass and a spherometer. The latter consists of a
tripod, with a micrometer screw in the centre, whose point may be
moved to any desired distance above or below the plane of the
three legs on which it rests. The most important qualities are
lightness and stiffness, and on this account a very cheap, and quite
efficient spherometer may be made with the nut and tripod of
wood, using for legs, pieces of knitting needles.
Experiment. Stand the spherometer on the sheet of plate glass
and turn the screw until its point is in contact with it. There are
three ways of determining the exact position of contact. The first
method is dependent on the fact that if the point of the screw is
too low the spherometer will stand unsteadily, like a table with
one leg too short. The screw is therefore depressed until the in-
strument rattles, when its top is moved gently from side to side.
An exceedingly small motion of this kind is perceptible to the
hand. The screw is then turned up and down until the exact
point of contact is found. The second, and probably the best
method, is to turn the screw slowly, taking care that no greater
pressure is exerted on one leg than on the other ; as soon as the
point touches the glass the pressure is removed from the legs, and
the friction of the nut at once makes the whole instrument revolve.
Care must be taken not to press on the top of the screw, or the
tripod will be bent, and an incorrect reading obtained. The
third method of determining contact depends on the sound pro-
SPHEROMETER. 43
duced when the instrument slides over the glass, which changes
when the screw touches the surface. It should be moved but a
short distance and without pressure, for fear of scratching the glass.
Having determined this point with accuracy, read the position
of the screw, taking the number of revolutions from the index on
one side, and the fraction from the divided circle.
Place the spherometer on each face of the two lenses and meas-
ure the position of the point of contact as before. Of course the
screw must be raised when the surface is convex, and depressed
when it is concave. Subtract each of these readings from that
taken on the plate glass, and the difference gives the height of a
segment of the sphere to be measured, whose base is a circle pass-
ing through the three feet of the spherometer. Call this height h
the radius of the circle r, and the radius of
the sphere H ; then we have, Fig. 13, AB
= h, BD = r, and A C — E. But by sim-
ilar triangles AB : DB = DB : BE, or
r2 h
h : r = r : 2 J2 — A, or H = ^r + -5- Com-
—ii A
pute in this way the radius of each surface
of the lenses, remembering that a negative r. r
radius denotes a concave surface. To de-
termine r, measure the distance of each leg of the spherometer
from the axis of the screw, and take their mean. Measure also the
distances of the three legs from each other and take their mean.
They form the three sides of an equilateral triangle ; compute by
geometry the radius of the circumscribed circle, and see if this
value of r agrees with that previously found. Both r and h must
be taken in the same unit, as millimetres or inches, and great care
should be taken to make no mistake in the position of the decimal
point. The reduction of h is effected by multiplying it by the pitch
of the screw.
Finally, compute the principal focal distance, F, by the formula
-^ = (n — 1) I -^ -j- -vp , in which H and Rr are the radii of
the two surfaces, as computed above, and n the index of refraction
of the glass. The latter varies in different specimens, but in com-
mon lenses is about 1.53.
44 RATING CHRONOMETERS.
15. ESTIMATION OP TENTHS OP A SECOND.
Apparatus. A heavy body carrying a small vertical mirror is
suspended by a wire, so that it will swing by torsion, about once
in half a minute. A small telescope with cross hairs in its eye-
piece, is pointed towards the mirror, and a plate with a pin hole in
it is placed in such a position that when the mirror swings, the
image of the hole will pass slowly across the field of view of the
telescope, like a star. It may be made bright by placing a mirror
behind it and reflecting the light of the window. The whole ap-
paratus should be enclosed so as to cut off stray light. A good
clock beating seconds is also needed.
Experiment. Twist the mirror slightly, so that it shall turn
slowly. On looking through the telescope a point of light or star
will be seen to cross the field of view, at equal intervals of about
half a minute. Note the hour and minute, and as the star ap-
proaches the vertical line take the seconds from the clock and
count the ticks of the pendulum. Fix the eye on the star and
note its position the second before, and that after, it passes the
wire. Subdividing the interval by the eye we may estimate the
true time of transit within a tenth of a second. Take twenty or
thirty such observations and write them in a column, and in a sec-
ond column give their first differences. ' Take their mean and
compute the probable error. It will show how accurately you can
estimate these fractions of seconds.
This is called the eye and ear method of taking transits, which
form the basis of our knowledge of almost all the motions of the
heavenly bodies. It is still much used abroad, although in this
country superseded in a great measure by the electric chronograph
described on p. 16.
16. RATING CHRONOMETERS.
Apparatus. Two timekeepers giving seconds, one, which may
be the laboratory clock, to be taken as a standard, and a second to
be compared with it. For the latter a cheap watch may be kept
expressly for the purpose, or the student may use his own. If the
true time is also to be obtained, a transit or sextant is needed in
addition.
Experiment. First, to obtain the true time. As this problem
belongs to astronomy rather than physics, a brief description only
RATING CHRONOMETERS. 45
will be given. It may be done in two ways, with a transit or a
sextant ; the former being used in astronomical observations, the
latter at sea. A transit is a telescope, mounted so that it will
move only in the meridian. With it note by the clock the
minute and second when the eastern and western edges of the sun
cross its vertical wire, and take their mean. Correct this by the
amount that the sun is slow or fast, as given in the Nautical Al-
manac, and we have the instant of true noon. The interval be-
tween this and twelve, as given by the clock, is the error of the
latter.
The sextant may be used at any time when the sun is not too
near either the meridian or the horizon. A vessel containing mer-
cury is used, called an artificial horizon, and the distance between
the sun and its image in this is measured. Since the surface of the
mercury is perfectly horizontal, this distance evidently equals ex-
actly twice the sun's altitude. If the observation is made in the
morning, when the sun is ascending, the sextant is set at somewhat
too great an angle, if after noon at too small an angle, and the
precise instant when the two images touch is noted by the clock.
The sun's altitude, after allowing for its diameter, is thus obtained.
We then have a spherical triangle, formed by the zenith Z, the
pole jP, and the sun &. In this, PZ is given, being the comple-
ment of the latitude ; PS, the sun's north polar distance, is ob-
tained from the Nautical Almanac, and ZS is the complement of
the altitude just measured. From these data we can compute the
angle ZPS, which corrected as before and reduced to hours, min-
utes and seconds, gives the time before or after noon. The practi-
cal directions for doing this will be found given in full in J?ow-
ditches Navigator.
By these methods we obtain the mean solar time, which is that
used in every day life. For astronomical purposes sidereal time,
or that given by the apparent motion of the stars, is preferable.
It is found by similar methods, using a star instead of the sun.
In an astronomical observatory it is found best not to attempt to
make the clock keep perfect time, but only to make sure that its
rate, or the amount it gains or loses per day, shall be as nearly as
possible constant. We can then compute the error E at any given
time very easily by the formula E= Ef + tr, in which E' was the
46 . MAKING WEIGHTS.
error t days ago, and r the rate. By transposing we may also ob-
tain r, when we know the errors E and E', at two times separated
by an interval t. Take the last, two observations of the clock-
error, which should be recorded in a book kept for the purpose,
and compute the error at the time of your observation, and see
how it agrees with your measurement.
If the day is cloudy, or no instruments are provided for determ-
ining the true time, the experiment may be performed as follows.
Compute, as above, the rate and error of the clock. Next take
the difference in minutes and seconds between the clock and the
watch to be compared. To obtain the exact interval, a few sec-
onds before the beginning of the minute by the watch, note the
time given by the clock, and begin counting seconds by the ticks
of the pendulum. Then fixing your eyes on the watch, mark the
number counted when the seconds' hand is at zero. Repeat two
or three times, until you get the interval within a single second.
Now correcting this by the error of the clock, taking care to give
the proper signs, we get the error of the watch. The next thing
is to set the watch so that it shall be correct within a second. For
this purpose it must be stopped, by opening it and touching the
rim of the balance wheel very carefully with a piece of paper, or
other similar object. Set the minute hand a few minutes ahead to
allow for the following computation. Subtract the clock-error from
the time now given by the watch. It will give the time by the
clock, at which if the watch is started it will be exactly right. A
few seconds before this time hold the watch horizontally, with the
fingers around the rim, and at the precise second turn to the right
and then back. The impulse starts the balance-wheel, and the
watch will now go, differing from the clock by an amount just
equal to the error of the latter.
17. MAKING WEIGHTS.
Apparatus. A very delicate balance and set of weights, some
sheet metal, a pair of scissors, a millimetre scale, and a small piece
of brass, A, weighing about 18.4 grammes. The weights are best
made of platinum and aluminium foil ; but where expense is a
consideration, sheet brass may be used for the heavier, and tin foil
for the lighter weights. To improve the appearance of the brass
and prevent its rusting it may be tinned, or dipped in a silvering
PROPER METHOD OF WEIGHING. 47
solution, or perhaps better still, coated with nickel. Some steel
punches for marking the numbers 0, 1, 2 and 5, a mallet and sheet
of lead should also be provided.
PROPER METHOD OF WEIGHING.
A good balance is so delicate an instrument that the utmost care
is needed in using it. The student should thoroughly understand
its principle, and know how to test) both its accuracy and delicacy.
See Measurement of Weights, p. 19. The beam should never be
left resting on its knife-edges, or they will become dulled. It is
therefore commonly made so that it may be lifted off of them by
turning a milled head in front of the balance. A second milled
head is also added to raise supports under each scale-pan. To
weigh any object the following plan must be pursued. " To see if
the balance is in good order, lower the supports under the scale-
pans, then those under the beam, by turning the two milled heads.
The long pointer attached to the beam should now swing very
slowly from side to side, and finally come to rest at the zero. Re-
place the supports, and open the glass case which protects the bal-
ance from currents of air. The object to be weighed, if metallic
and perfectly dry, may be placed directly on the scale-pan, other-
wise it should be weighed in a watch-glass whose weight is after-
wards determined separately. Now place one of the weights in
the opposite scale-pan, and remove the supports first from the pans
and then from the beam. This must be done very slowly and
carefully. Students are liable to let the beam fall with a jerk
on the knife-edges, by which the latter are soon dulled and ruined.
An accurate weighing is necessarily a slow process and should
never be attempted when one is in a hurry. Moreover, by re-
moving the supports quickly the scale-pans are set swinging, and
the beam itself vibrating through a large arc, so that it will
not come to rest for a long time. It is better while using the
larger weights to lower the supports a very small amount only,
and notice which way the index moves. As it is below the beam
it always moves towards the lighter side. The smaller weights
must be touched only with a pair of forceps, as the moisture of the
fingers would soon rust them. Those over 100 grms. may be taken
up in the hand by the knob, but no other part of them should be
48 PROPER METHOD OF WEIGHING.
touched. Weights should never be laid down except on the scale-
pans, or in their places in the box. Now try weighing the piece
of brass A. Lay it on one scale-pan, and a 10 gr. weight on the
opposite side. The index moves towards the latter when the sup-
ports are removed, as described above. Replace the 10 grs. by
20 grs. This is too heavy, and the index moves the other way.
Try the 10 grs. and 5 grs. -too light; add 2 grs. -still too light;
another 2 grs. -too heavy; replace the latter by 1 gr. — too light.
The weight evidently lies between 18 and 19 grammes. Add the
.5 gr., or 500 mgr. -too heavy; substitute the 200 mgr.-too
light, and so go on, always following the rule of taking the weights
in the order of their sizes, and never adding small weights by
guess, or much time will be lost. Having determined the weight
within .01 gr., the milligrammes are most easily found by a rider.
This consists of a small wire whose weight is just 10 mgr. It is
placed on different parts of the beam, which is divided like a steel-
yard into ten equal parts, which represent milligrammes. Thus if
the rider is placed at the point marked 6, or at a distance of .6
the length of one arm of the balance, it produces the same effect
as if 6 mgrs. were placed in the scale-pan. It is generally arranged
so that it can be moved along the beam without opening the glass
case, which protects the latter from dust and currents of air. By
taking care to lower the supports of the beam slowly, as recom-
mended above, the swing of the index is made very small ; it is
sufficient to see if it moves an equal distance on each side of the
zero, instead of waiting for it to come absolutely to rest. To make
sure that no errror is made in coimting the weights, their sum
should be taken as they lie in the scale-pan, and also from their
vacant places in the box.
Decimal weights are made in the ratio of 1 , 2 and 5, and their
multiples by 10, and its powers. To obtain the 4 and the 9 it is
necessary to duplicate either the 1 or the 2. The English adopt
the former method, the French the latter. Comparing the two
mathematically, we find that using the weights 5, 2, 2, 1, we shall,
on an average in ten weighings, remove a weight from box to
scale-pan 34 times, of which it will be put back 17 times during
the weighing, and the remaining 17 times after the weighing is
completed. In the English method, with the weights 5, 2, 1, 1,
PROPER METHOD OF WEIGHING. 49
under the same circumstances the weights are again used 34 times,
replaced 15 times during the weighing, and 19 after it. There is
therefore no difference in rapidity of one plan over the other. The
French system has, however, the great advantage that we may at
any time test our weights against one another, since 1 -j- 2 -f- 2
should equal the 5 weight, and sometimes in weighing, if a mis-
take is suspected a test may be applied by using the additional
weight instead of putting back all the small weights, and adding a
larger one, as is necessary in the English system. To meet this
difficulty a third 1 gramme weight is sometimes added by English
makers.
Experiment. To make a set of weights for weighing fractions
of a gramme. Four are needed of platinum or brass weighing
500, 200, 200 and 100 mgrs., and four of aluminum, or thick tin
foil, weighing 50, 20, 20 and 10 mgrs. The latter should be made
first, since being the lightest they are the easiest to adjust. Cut a
rectangle of the foil about 3 or 4 centimetres on a side, and weigh
it within a milligramme. Now determine its area by measuring its
four sides and taking the product of its length by its breadth. If
the opposite sides are not equal, take their mean. Let A equal
the area, and TFthe weight of the foil. Evidently W -f- A will
equal t0, the weight per square millimetre, and 50, 20 and 10 di-
vided \)jw will give the areas of the required weights. Cut pieces
somewhat too large and reduce them to the proper size by the
Method of Successive Corrections, p. 10. This is accomplished by
weighing each and dividing its excess by w. The quotient shows
how much should be cut off. As they cannot easily be enlarged
if made too small, and the thickness of the foil may not be the
same throughout, pieces should be cut off smaller than the com-
puted excess. Small amounts may be taken from the corners, and
when completed one of the latter should be turned up to make- it
easier to pick them up with the forceps. Finally, lay them on the
plate of lead, and stamp their weight in milligrammes on them, with
the steel punches and mallet. Do the same with the heavier foil,
thus making the 500, 200, 200 and 100 mgrs. weight. More care
is needed with them, and the last part of the reduction should be
effected with a file. Unless great care is taken, two or three will
50
DECANTING GASES.
be spoiled by making them too light, before one of the right
weight is obtained.
18. DECANTING GASES.
Apparatus. A pneumatic trough, which is best made of wood
lined with lead, and painted over with paraffine varnish. A gradu-
ated glass tube Z>, Fig. 14, closed at one end, and holding about
100 cm.3, a tubulated bell-glass B containing about a litre, with
stop-cock G attached, and two or three dry Florence flasks. The
mouths of the latter should be ground, so that they may be closed
by a plate of ground glass ; to remove the moisture they should be
heated in a large sand bath, or over steam pipes. A thermometer
is also needed.
Experiment. Measure the temperature of the air in the flask A
by the thermometer, also its moisture, or rather its dew-point. The
latter may be assumed to
be the same as that of the
room, and obtained from
the student, using the me-
teorological instruments.
Now close the flask with
the plate of glass, and im-
merse it neck downwards
in the pneumatic trough.
It may be kept in this po-
sition for any length of time, as the water prevents the air from
escaping. Next fill the large graduated vessel J3 with water, by
opening its stop-cock C and immersing, then close C and raise
it. Now decant the gas into it by pouring, just as you would pour
water, only that it ascends instead of falling. When all has been
transferred read very carefully the volume, as given by the gradu-
ation, also the approximate height of the water inside above that
outside the jar. Dividing this difference by 13.6 the specific grav-
ity of mercury, and subtracting the quotient from the height of the
barometer, gives the pressure- to which the enclosed air is subjected.
Its temperature may be assumed equal to that of the water, and it
may be regarded as saturated with moisture. Next, to transfer it
into the graduated tube 7>, attach a rubber tube to (7, and after fill-
REDUCTION OF GASES. 51
ing D with water and inverting it in the trough, let the air bubble
into it from the tube by opening G and lowering JB. When D is
nearly full, close (7 so as to prevent the escape of the air, and read
and record the volume as given by the graduation on D. Now
decant the air from J) into the flask A. Great care is necessary in
this operation to prevent spilling, and it is best to practise a few
times beforehand, until it can be transferred without allowing a
single bubble to escape. Continue to empty B until all the air has
been passed into A. The latter will then be nearly full of air,
unless some has been lost. In the latter case do not give up the
experiment, but keep on, retaining as much air as possible. Now
holding the neck of the flask in the hand press the ground glass
against it with the thumb, so as to retain what water is still in it,
and taking it out of the trough stand it on the table right side up.
Wipe the outside dry, and weigh it in its present condition ; also
when full of water, and when empty. Call the three weights m,
n and o, respectively. The volume of air in cm.8, at the beginning
of the experiment, will equal n — o in grammes ; that at the end
n — m. There are now four volumes of air to be compared. First
the volume at the beginning of the experiment, when the air was
moist and the dew-point was given ; secondly, when transferred
to £ ; thirdly, that found by adding the readings of D ; and
fourthly, that at the close of the experiment. Reduce all of these
to the standard pressure and temperature by the method given
below, when they should be equal if no air has escaped, other-
wise the difference shows the amount of the loss. Great accuracy
must not be expected, owing to the absorption of the air by the
water, and for various other reasons.
REDUCTION OF GASES TO STANDARD TEMPERATURE AND
PRESSURE.
I. Dry Gas. Given a volume T^P of dry gas at temperature £,
and barometric pressure P, to find what would be its volume T^H
if cooled to 0° C, and the pressure altered to the standard H =
760 m.m. Suppose that it is first cooled to 0°, without changing
the pressure, and call its new volume T^P. We have by Gay
Lussac's law for the expansion of gases, FJP = T^P (1 + «£), m
which a = 2Tj, the coefficient of expansion of gas. Again, by
52 STANDARDS OF VOLUME.
Mariotte'e law we have, VOF : Vm = JET : P. Hence Vof =
JET H
VoH'-> or substituting, VtP = Ym-(l + «0*J or»
97.Q p
C\ ^
_
tp (273 + <) 760
For any other temperature £', and pressure JP', we have,
273 Pr T_ P 273 + H
FOH = Y«* (273 + 0 760 ' hence F*^ = Fflp P7' 273 +«' '
The first formula is used to determine the true quantity of gas
present, that is, the volume at the standard temperature and pres-
sure. The second, to compute the new volume when we alter
both temperature and pressure.
II. Gas saturated with Moisture. Call p the pressure of
aqueous vapor at the temperature t. Then of the total pressure P
we have p due to the vapor, and P — p to the gas ; substitute?
therefore, P — p for P in equation (1), and we have,
-^)
III. Gas moist, but not saturated. Let the gas be gradually
cooled, until the temperature becomes so low that the moisture
can no longer be retained as vapor, but begins to condense on the
walls of the vessel. This temperature T is called the dew-point;
let p' be the corresponding pressure of the vapor. Then p : p' =
1 +at : 1 + aT, or p = p'(\ + at) -T- (1 + aT), and substitut-
ing this value in equation (3^), we have
273 r (1 + at) n
y°R ~ Ktp (273 + t) 760 ' Lr " p (1 + a T)J '
19. STANDARDS OF VOLUME.
Apparatus. A balance AB, Fig. 15, capable of sustaining 5
kgrs. on each side, and turning with a tenth of a gramme under
this load. Remarkably good results may be obtained with com-
mon balances, such as are used for commercial purposes, by attach-
ing a long index to the beam, as in the figure. Several pounds of
distilled water should be provided, a thermometer, a set of weights,
and a rubber tube and funnel. Instead of a scale-pan, a counter-
poise C is attached to one arm of the balance as a method of
double weighing is to be used. The standard to be graduated,
which we will suppose to be a tenth of a cubic foot, consists of a
glass vessel D, whose capacity somewhat exceeds this amount. A
STANDARDS OF VOLUME.
53
J" steam valv.e is screwed into the cap closing the lower end
which also carries a sharp brass point to form the lower limit of
the volume. A ring is attached to the cap closing the upper end
of the vessel, by which the whole is supported. A brass hook
with the point turned upwards passes through this cap, in which
a hole has been drilled to allow the air to pass in or out. The
hook may be raised or lowered, and clamped at any height by a
conical nut surrounding it, or by a set screw. Finally a millimetre
scale should be attached to the upper end of D.
Experiment. Note the height of the barometer, the temperature
of the room, also that of the distilled water. Fill D, by attaching
the rubber tube, as in
the figure, opening E
and pouring in the
water. When the ves-
sel is full, close E
and remove the rubber
tube. Take care that
no air bubbles adhere
to the side of the H
glass. • Open E and
Fig. 15.
draw off the water until it stands just on a
level with the top of the scale attached to
the glass. Counterpoise by adding weights to
the scale-pan F, until the index stands at zero,
first reading the directions for weighing, given
on page 47. t)raw off enough water to lower
its level just one centimetre, counterpoise
again, and repeat until the surface reaches the
bottom of the scale. If too much water is removed at any time
refill the vessel above the mark, and draw off the water again.
Now bring the water level just above the point of the hook, and
close E, so that the flow shall take place drop by drop. Use the
hook as in Experiment 13, and as soon as the point becomes
visible close E. Read the level of the water and counterpoise as
before. Repeat two or three times, adding a little water after each
measurement. Now open JE7, and let the water run out until the
lower point just touches the surface. Measure the temperature of
the water as it escapes. To counterpoise the beam nearly three
54 STANDARDS OF VOLUME.
kilogrammes additional must be added to F. Make this weigh-
ing with care, and repeat two or three times, as when observing
the upper point. Subtract each of the weights when the vessel
was full, from the mean of those last taken, and the difference
gives the weight of the water contained between the lower point
and each of the other observed levels.
Now to determine the volume, we have given by Kater, the
weight of 1 cubic inch of distilled water at 62° F., and 30 inches
pressure, equals 252.456 grains, and 1 gramme equals 15.432 grains.
From this compute the weight of one tenth of a cubic foot. Two
corrections must now be applied, the first for temperature, the
second for pressure. Water has an expansion of about .00009
per 1° F. when near 62°, and glass .000008 linear, or three times
this amount of cubical expansion at the same temperature; of
course the apparent change of volume is the difference of in-
crease of the water, and of the glass. Evidently at a high
temperature less water would be required, hence this correction is
negative if the temperature is above 62°. Practically in making
standards it is best to keep the temperature exactly at 62°, adding
ice or warm water if necessary, as this correction is a little doubt-
ful, owing to the unequal expansion of different specimens of glass.
The vessel D is buoyed up by the air, by an amount equal to the
weight displaced, and this weight is evidently proportional to the
barometric pressure H. Now 100 cubic inches of air at 30 inches
weigh 2.1 grms., hence at 1 inch it would be f^, and if the pres-
sure is changed from 30 to H, the change in weight would evi-
dently be 2.1 X (30 — H) 4- 30. The weights, however, are also
buoyed up in the same way, but as the specific gravity of brass
is about 8, the effect is only one-eighth as great. The true
correction is then seven-eighths of this amount. The higher the
barometer the greater the buoyancy, and the lighter the water
will appear, or this correction will be negative for pressures above
30 inches. Both the corrections will be small, and in most cases
can be neglected ; but it is well to make them, in order to be sure
to understand the principle. Having thus computed how much
the tenth of a cubic foot ought to weigh, see if the distance be-
tween the points is correct, and if not, determine by interpolation
BEADING MICROSCOPES. 55
where the water level should be in order to render the capacity
exact.
20. READING MICROSCOPES.
Apparatus. Three cheap French microscopes mounted on
moveable stands, as in AB, Fig. 16. Two should have cross-hairs
in their eye-pieces, while the third should contain a thin plate of
glass with a very fine scale ruled on it. An accurate scale divided
into millimetres is required as a standard of comparison, and since
the division marks of those in common use are too broad for exact
measurements, it is better to have one made to order, with very
fine lines cut on the centre of one face instead of on the edge.
The best material is glass, but copper or steel will do, especially if
coated with nickel or silver. Several objects to be measured
should be selected, as a rod pointed at each end, the two needle
points of a beam-compass, and a scale divided into tenths of an
inch, whose correctness is to be tested. Under the microscopes is
placed a board D, on which the object to be measured (7, is laid,
and which may be raised or lowered gradually by screws, or fold-
ing wedges. Another method of supporting the microscopes,
superior in some respects, will be found described under the Ex-
periment of Dilatation of Solids by Heat.
Experiment. If a measurement within a tenth of a millimetre
is sufficiently exact, use the two microscopes with cross-hairs.
Place them at such a distance apart that each
shall be over the end- of the object to be meas-
ured, which should be laid on D. They
should be raised or lowered until in focus, and
then set so that their cross-hairs shall exactly
coincide with the two given points. Remove Fis- 16-
the object very carefully, so as not to disturb their position, and
replace it by the standard scale, bringing the zero to coincide
with one of the cross-hairs. Now looking through the other
microscope read the position of its cross-hairs on the scale, esti-
mating the fractions of a millimetre in tenths. If the image of
the scale is not distinct it may be focussed by slowly raising or
lowering the board on which it is placed, taking great care not
to disturb the microscopes. To get the whole number of milli-
metres, a needle may be laid down on the scale, and the right
division distinguished by its point.
If greater accuracy is desired, use the third microscope, find-
56 DIVIDING ENGINE.
ing the magnitude of the divisions of its scale in the following
manner; focus it on the steel scale, placing it so that two divi-
sions of the latter shall be in the field at the same time. Read
each of them by the scale in the eye-piece, and take the differ-
ence ; the reciprocal is the magnitude of one division in millimetres.
Repeat a number of times and take the mean. To make any
measurement, place this microscope with one of the others over the
points to be determined, and take the reading with its scale, esti-
mating tenths of a division ; then substitute the steel scale as be-
fore, and read the millimetre mark preceding, also that following.
By a simple interpolation the distance is obtained from these three
readings with great accuracy.
Try both these methods with the objects to be measured, and
then test the scale of tenths of an inch by measuring the distance
of each inch mark from the zero, and reducing the millimetres to
inches. Measure also in the same way the ten divisions of one of
the inches.
One of the best ways to measure off a large distance, as ten or
twenty metres, with accuracy, is by means of a couple of reading
microscopes. A steel rule is used, the ends being marked by the
microscopes, as they are in rough measurements, by the finger. In
all cases where the graduation extends to the end of the rule it is
better to use the mark next to it, both as being more accurate, and
as affording a better object to focus on.
21. DIVIDING ENGINE.
Apparatus. This instrument rests on a substantial stand
ABED, Fig. 17, like the bed-plate of a lathe. A carefully con-
structed micrometer screw moves in this, and pushes a nut y from
end to end. The screw should have a pitch of about a millimetre,
or a twentieth of an inch, if English measures are preferred. The
head of the screw is divided into one hundred parts, and turns
past an index which is again divided into ten parts, as in Fig. 4,
]). 24. The screw may be turned by a milled head or a crank.
The nut must have, a bearing of considerable length, a decimetre
is scarcely too much, as any irregularities are thus compensated.
It should be split so that it may be tightened by screws, or better,
by a spring, and slides along two guides, AB formed like an in-
verted V, and DE, which is flat. A scale is cut on the latter to
give the whole number of revolutions of the screw. The nut
DIVIDING ENGINE. 57
should move with perfect smoothness from end to end, but not too
freely. A certain amount of bac7c-lash is unavoidable (that is, the
screw may always be turned a short distance backwards or for-
wards without moving the nut), but this does no harm, as when
in use it should always be moved in the same direction. A second
screw similar to the other, but smaller, and at right angles to it, is
attached to C, so that its nut may be moved backwards or for-
wards about one decimetre. It carries a reading microscope J?,
made of a piece of light brass tubing, by inserting an eye-piece
above, and screwing a microscope objective into the lower end.
It may be focussed by sliding the tube up and down by a rack
and pinion. Cross-hairs should be placed in the eye-piece, but
in some cases a fine scale, or eye-piece micrometer, is preferable.
To use this instrument as a dividing engine, the microscope must
be made movable, so that it can be replaced by a graver for metals,
or a pen for paper. The micrometer head F has ten equidistant
holes cut in it, in which steel pins can be inserted. These strike
against a stop which they cannot pass unless it is pushed down by
the finger. A sheet of thick plate glass DSTE serves as a stand
on which to lay objects, and under it is a large mirror to illuminate
them, but it may be removed when desired.
Experiment. This instrument may be applied to a great variety
of purposes. Several experiments with it will therefore be de-
scribed.
1st. To test the screw. Lay a glass plate divided into tenths
of a millimetre on DSTE,
and bring the microscope
over it. Use a moderately
high power, as a J" objective,
and focus on the scale; the
want of a fine adjustment
may be partly remedied by
varying the distance of the
eye-piece from the objective.
Bring the first division of the
scale to coincide with the
cross-hairs of the microscope
by turning the micrometer-
head F. Read the whole number of turns from the scale on DE,
and the fraction from F. Move it one or two turns to the right,
and set again ; repeat several times, and compute the probable error
58 DIVIDING ENGINE.
of one observation. It equals the error of setting. Turn the
screw the other way, and bring it back to the line. The differ-
ence between this reading or the mean of ten such readings, and
that previously obtained, gives the back-lash. Set in turn on sev-
eral successive points of the scale. The first differences should be
equal. Mark two crosses on a plate of glass with a diamond, three
or four centimetres apart. Measure the interval between them
with different portions of the screw, and see if they agree. If not,
the defect in the screw must be carefully examined, and corrections
computed. The screw M should be similarly tested.
2d. Determination of the pitch of the screw. Procure a
standard decimetre (or other measure of length) and measure the
distance between its ends. The temperature should be nearly that
taken as a standard, or if great accuracy is required, allowance made
for the difference of expansion of the screw and decimetre. From
this measurement, which should be repeated several times, compute
the true pitch of the screw, and the correction which must be ap-
plied when distances are measured with it.
3d. To measure any distance. Lny the object on the glass
plate and bring the cross-hairs of the microscope to coincide first
with one end of it, and then with the other. The difference in
the readings is the length. Apply to it the correction previously
determined.
4th. To determine the form of any curved line. For exam-
ple, use one of the curves drawn by a tuning fork, in the Experi-
ment on Acoustic Curves. Bring the cross-hairs to coincide with
several points in turn of one of the sinuosities, and read both
micrometer heads. These give two coordinates, from which the
points of the curve may be constructed on a large scale, and com-
pared with the curve of sines, the form given by theory. The
relative positions of a number of detached points may also be thus
determined, as in the photographs of the Pleiades and other
groups of stars by Mr. Rutherford.
5th. Graduation. For a first attempt, make a scale on paper
with a pencil or pen. Replace the microscope by a hard pencil
with a flat, but very sharp point. It must be arranged so that it
can be moved backwards or forwards a limited distance, but not
sideways. Every fifth line should be longer than the rest, which
RULING SCALES. 59
should be exactly equal to each other in length. Fasten the paper
securely on the glass plate so that it shall not slip. Suppose now
lines are to be drawn at intervals of half a millimetre. Insert a
pin in one of the holes in F, and turn the latter to the stop.
Draw a line with the pencil for the beginning of the scale, depress
the stop to let the pin pass, give F one turn, bring the pin again
to the stop and draw a second line, and so on. If the lines are to
be a millimetre apart, draw one line for every two turns. In the
same way, by inserting more pins a finer graduation may be ob-
tained. Instead of using the pins a table may be computed
beforehand, giving the reading of the screw for each line to be
drawn, allowing for the errors of the screw, if great accuracy is re-
quired. The scale is then ruled by bringing the nut successively
into the various positions marked in the table, and drawing a line
after each.
A most important application of this instrument is to the meas-
urement of photographs of the sun taken during eclipses. The
position of the moon at any instant is thus obtained, with a degree
of precision otherwise unattainable. In this, and other cases
where angles must also be measured, the plate of glass ES should
be removed, and the object laid on a rotary stand, with a gradu-
ation showing the angle through which it is turned.
22. RULING SCALES.
Apparatus. In Fig. 18, two strips of wood A and J?, rest on a
smooth board, and are held in place by the weights C and D.
The ends of a string are attached to them, which is stretched by
means of a weight F, so that if C and D are raised A and J5 will
slide. A peg is inserted in B, which moves between two steel
plates fastened to A, one being fixed, the other movable by means
of a screw Gr. If, then, either weight is raised, the strip of wood
on which it rests will be drawn forward by F, but will be free to
move through a space equal to the difference of the diameter of
the peg and the interval between the two steel plates. If desired,
Gr may be a micrometer screw, by which this interval may always
be accurately determined. It may be fastened in any position by
a clamp or set screw. A steel rod H is used to draw the division
lines. It is fixed at one end, and carries at the other a pencil, pen,
graver or diamond, according as the lines are to be drawn on pa-
per, metal or glass. By this arrangement there is little or no
60 RULING SCALES.
lateral motion of the graver, but unfortunately it draws a curved
line. To remedy this defect, the rod may be replaced by a stretched
wire, to the centre of which the graver is attached, or the latter
may slide past a guide against which it is pressed by a spring.
Experiment. For many purposes in using a scale, it makes but
little difference what the divisions are, provided that they are all
equal, and this is especially the case in all accurate measurements,
since as a correction must always be made for temperature, we
can readily at the same time correct for the size of the divisions.
The instrument here described will probably give divisions more
nearly equal than those obtained by a micrometer screw, but it is
more difficult to make them of any exact magnitude, since any
deviation is multiplied by the number of divisions.
To draw a scale, lay a piece of paper on B and fasten it with
tacks or clips. To secure uniformity in the length of the long and
short division marks, rule three
parallel lines as limits, attach
a sharp flat-pointed pencil to'
H, and slide A and B until
the beginning of the scale
is under JET. Draw a line with
the latter, and make one stroke
with the machine. This is done
by raising (7, when F will draw
A forward a distance equal to the interval between the two plates
near Gr, minus the thickness of the peg. Lay C down and raise
D. A will now remain at rest, but B will move through the same
distance. Draw a second line with the pencil, and repeat, making
every fifth line about twice as long as the others. They will be
found spaced at distances which may be regulated by the screw Gr.
Try making short scales in the same way, with large and small
divisions. It is always safer to keep the hand on one weight while
the other is lifted. The magnitude of F should be such that the
strajn on the cord will be greater than the friction of repose when
the weights are up, but less than the friction of motion when they
are down. If F is too light, when C is raised A will not start ;
if too heavy, it will strike so hard that it will move B. To test
the accuracy of the machine draw a single line, take a hundred
RULING SCALES. 61
strokes and draw another. Then without moving G- push the
slides back and draw a third line close to the first ; take a hundred
strokes and draw a fourth line near the second. Measure the in-
terval between the first and third, and the second and fourth.
They should be equal, but if not, the difference divided by an
hundred gives the average difference in length of a stroke the
second time, compared with the first.
Instead of a pencil, a pen may be used to draw the lines, or a
graver, if a metallic scale is desired. The finest scales are ruled
on glass by a diamond. Instead of using the natural edge of the
gem, as when cutting glass, an engraver's diamond should be em-
ployed, which is ground with a conical point ; the direction in
which it should be held, and the proper pressure, being obtained by
trial. Scales may also be etched by covering the surface with a
thin coating of wax or varnish, and the lines marked with a
graver. If metallic, it is then subjected to the action of nitric
acid ; if of glass, to the fumes of fluorhydric acid. It is possible
that the new method of cutting glass by a sand-blast may prove
applicable to this purpose with a great saving of time and trouble.
MECHANICS OF SOLIDS.
23. COMPOSITION OF FORCES.
Apparatus. Two pulleys A and .Z?, Fig. 19, are attached to a
board which is hung vertically against a wall. Two threads pass
orer them, and a third (7, is fastened to their ends at D. Three
forces may now be applied by attaching weights to the ends of
the cords. The weights of an Atwood's machine are of a con-
venient form, but links of a chain, picture hooks, cents, or any
objects of nearly equal weight may be used. Small beads are
attached to the three threads at distances of just a decimetre
from D.
Experiment. Attach weights 2, 3 and 4 to the three cords, and
let D assume its position of equilibrium. Owing to friction it will
remain at rest in various neighboring positions,
their centre being the true one. Now meas-
ure the distance of each bead from the other
two witn a millimetre scale, and obtain -the
angle directly from a table of chords. If
these are not at hand, dividing the distance
by two, gives the natural sine of one half the
required angle. By the law of the parallelo-
gram of forces, the latter are proportional to
the sides of a triangle having the directions of
the forces. But these sides are proportional to the sines of the
opposite angles, hence the sines of the angles included between the
threads should be proportional to the forces or weights applied.
Divide the two larger forces by the smaller, and do the same with
the sines of the angles, and see if the ratios are the same. The
angles themselves should first be tested by taking their sum, which
should equal 360°. If either angle is nearly 180°, it cannot be
accurately measured in this way, but must be found by subtracting
the sum of the other two from 360°, or measuring one side from
Fig. 19.
MOMENTS. 63
the prolongation of the other. It is well to draw the forces from
the measurement, and see if a geometrical construction gives the
same result as that obtained by calculation. Repeat with forces in
several other ratios, as 3, 4, 5 ; 2, 2, 3 ; 3, 5, 7 ; taking care in all
cases to include in the weights the supports on which they rest.
24. MOMENTS.
Apparatus. A board AB, Fig. 20, is supported at its centre of
gravity on the pin O. It should revolve freely, and come to rest
in all positions equally. Two forces may be applied to it by
the weights D and E, attached by threads to the pins F and 6r.
Their magnitudes may be varied from 1 to 10 by different weights,
and their points of application by using different pins, as H, I and
J. To measure their perpendicular distances from the pin C, a
wooden right-angled triangle or square is provided, one edge of
which is divided into millimetres, or tenths of an inch.
^Experiment. Various laws of forces may be provred with this
apparatus. 1st. When a single force acts on a body AB fixed at one
point, as (7, there will be equilibrium only when it passes through
this point. Remove FD and attach a weight E to &. It will be
found that the body will remain at rest only when the point G is
in line with M and C. 2d. A force produces the same effect if
applied at any point along the line in which it tends to move the
body. Apply the two weights D and E, which tend to turn the
board in opposite directions. Make
their ratio such that MGr shall be in
line with G-, H, J. Now transfer the
end of the thread from G- to Jf, ./and
<7in turn, when it will be found that
the position of the board will be un-
changed. It should be noticed, how-
rig. 20.
ever, that in the last case the board is
in unstable equilibrium, since FJ falls beyond the point of support
C. 3d. The moment of a force, or its tendency to make a body
revolve, is proportional to the product of its magnitude by its per-
pendicular distance from the point of support. Make D equal 2,
and attach it to JT, so that the thread rests over the edge of the
board, which is the arc of a circle with centre at (7, and radius .6.
Its tendency to make the board revolve is therefore the same, what-
64
PARALLEL FORCES.
ever the position of the latter. Make E successively 1, 2, 3, 4, 5,
6, and measure the perpendicular distance of the thread to which
it is attached in each case from C. This distance is measured by
resting the triangle against the thread and measuring the distance
of 6rby its graduated edge. In each case the moment of E will be
found to be the same, and equal to 2 X 6, the moment of D. 4th.
When two forces hold the body in equilibrium their resultant must
pass through the fixed point. Make D equal 2, and attach it to
F, and E equal 3, applied at G. Lay a sheet of paper on the right
hand portion of AB, making holes for F, C and J~to pass. Draw
on it with a ruler the direction of the two threads prolonged, and
then removing it, construct their resultant geometrically by means
of the parallelogram of forces. It will be found to pass through
C. Repeat two or three times with different weights and points
of application.
25. PARALLEL FORCES.
Apparatus. The apparatus used is shown in Fig. 21. AJB is a
straight rod about two feet long, with a paper scale divided into
tenths of an inch attached to it. It is supported by a scale-beam
CD with a counterpoise, so that it is freely balanced, and remains
horizontal. Weights formed like those of a platform scale may be
attached to it at any point, by riders, as at E, F and O. Taking
each rider as unity, four sets of weights are required of magni-
tudes 10, 5, 2, 2, 1, .5, .2, .2, .1. Two other beams, like CD,
should also be provided, to which these weights may be attached,
as at E, so as to produce an upward force of any desired magni-
tude. All these scale-beams may be very roughly made, even
a piece of wood supported at the centre by a cord, being suf-
ficiently accurate. English beams of iron may, however, be ob-
tained at a very low price.
The
Fig. 21.
resultant of any system of parallel forces
lying in one plane may be found by
this apparatus. Thus suppose we
have a force of 15.7 acting upwards,
and two of 8.3 and 1.4 acting down-
wards, and distant from the first
6.4 and 8.7 inches respectively.
Produce the upward force by add-
ing the weights 14.7 to E, and the
PARALLEL FORCES. 65
two downward forces by weights 7.3 and .4 (allowing 1 for each
of the scale-pans) at F and G, setting them at the points of the
beam marked 3.6 and 18.7. They are then at the proper distance
from C, which is at 10 inches from the end. We now find that
A goes down and B up ; by placing the finger on the beam we
see that it can be balanced only by applying a downward force to
the right of C. Now place a rider in this position, and move it
backwards and forwards, varying the weight on it until the beam
is exactly balanced. The magnitude of this weight will be found
to be 6, and its position 16.8, or 6.8 inches from C. The resultant
of the three forces will be just equal and opposite to this. Had
the force required to balance them acted upwards, we should havo
used one of the auxiliary scale-beams. To test the correctness of
this result we compute the resultant thus : JR, = 15.7 — (8.3 -f-
1.4) = 6, and taking moments around C we have 8.3 X 6.4 — 1.4
X 8.7 = 6 X aj, or x = 6.8 the observed distances.
Determine the position and magnitude of the resultant in sev-
eral similar cases, as for example the following, in which IT means
an upward, and D an downward force, and each is followed first by
its magnitude, and then by the point on the bar at which it is to
be applied.
t D, 5.0, 4.3 ; Z7J 10.0, 10.0.
2. D, '2.6, 3.2 ; IT, 7.8, 10.0.
3. D, 7.4, 3.7 ; IT, 17.1, 10.0.
4. D, 11.1, 2.1 ; D, 6.5, 5.6; IT, 2.3, 18.4.
5. D, 5.2, 1.9; U, 15.2, 10.0 ; D, 8.4, 12.6; IT, 3.0, 18.1.
Two equal parallel forces acting in opposite directions and not
in the same line, form what is called a couple, and have no single
resultant. Thus apply the two forces D, 12.0, 5.0, and IT, 12.0, 10.0.
No single force will balance the beam. Equilibrium is obtained
only by a second couple having the same moment, and turning in
the opposite direction ; thus the moment being 12.0 X 5.0 = 60.0,
the beam may be balanced by two forces of 10.0, each distant 6
inches from one another, placing the upward force to the left.
Find in the same way some equivalent to D, 4.3, 7.6, and U, 4.3,
10.0, and notice that it makes no difference to what part of the
beam the two forces are applied, provided their distance apart
remains unchanged.
5
00 CENTRE OF GRAVITY.
This same apparatus may be applied to illustrate the case of a
body with one point fixed, acted on by parallel forces, as, for ex-
ample, the lever, by using a stand IT with two pins, between which
the beam may turn. This stand is also useful in finding the point
of application of the resultant in the above cases.
26. CENTRE OF GRAVITY.
Apparatus. Several four-sided pieces of cardboard (not rec-
tangles) and a plumb line, made by suspending a small leaden
weight by a thread, from a needle with sealing wax head.
Experiment. Make four holes in the cardboard, two AB, Fig.
22, close to two adjacent corners, the others in any other part not
too near the centre. Pass the needle through A and support the
cardboard by it ; the thread will hang vertically downwards, and
the centre of gravity must lie somewhere in this line, or it would
not be in equilibrium. Mark a point on this line as low down as
possible, and connect it with the pin hole. Do the same with B ;
the intersection of the two is the centre of gravity. Turn the
cardboard over and repeat with the other holes. This gives two
determinations of the centre of gravity. To see if the two points
are opposite one another, prick through one and see if the hole coin-
cides with the other. By suspending at any other points, the same
result should be obtained. Be careful that the holes are large
enough to enable the card to swing freely.
Next, lay the card down on your note book and mark the four
points A, JB, C, .D. Connecting them with lines gives a duplicate of
the cardboard. On this construct the centre of
gravity geometrically. Divide into two trian-
gles by connecting A O. Bisect AD in E, and
CD in F. The centre of gravity of A CD
must lie in AF, also in CE^ hence at G. Ob-
tain Q-' by a similar construction with AB C.
The centre of gravity of the whole figure must
lie in GG-'. Make a second construction by
connecting BD, making the triangles ABD and B CD ; the in-
tersection of QGr and its corresponding line gives the centre
of gravity. Lay the piece of cardboard on the figure and prick
CATENARY.
67
through the two centres of gravity previously found. They should
agree closely with that found geometrically.
27. CATENARY.
Apparatus. A chain three or four yards long, each link of which
is a sphere, known in the trade as a ball link chain. Every tenth
link should be painted black, and the fiftieths red. A horizontal
scale A B (7, Fig. 23, attached to the wall, also a number of pins to
which the chain may be fastened by short wire hooks, and its
length altered at will. A graduated rod BD is used to measure
the vertical height of any point of the chain.
_O— |-<-H— 1~- H-+-T
Experiment. First, to determine the average length of the
links. Let the chain hang vertically from A, measure the length
of each hundred links, and take
their mean. A simple proportion
gives the number of links to
which AC is equal. Suspend
the chain at A and (7, making the
flexure at the centre about half a
foot. Measure it exactly, and in-
crease the original length 10 links
at a time to 100. Increase it also
by 17 links, by 63 and by 48, and
measure as before. Write the re-
Fig. 23.
suits in a column and take the first, second and third differences
of the first measurements. Now obtain by interpolation the three
values for 17, 63 and 48 links, and compare with their measured
values.
Next suspend the chain as in ADE, and measure the deflection
at intervals of five inches horizontally. This is best done by pass-
ing a pin through the graduated rod at the zero point, letting it
hang vertically, then measuring by it. Taking differences as be-
fore, those of the first order will be at first negative, then increase
until they become positive. Where the first difference is zero,
is evidently the lowest point of the curve. By the method of
inverse interpolation find this point, treating the first differences
as if they were the original variable, and recollecting that each
difference belongs approximately to the point midway between the
68 CRANK MOTION.
two terms from which it was obtained. Thus the difference ob-
tained from the 5 and 10 inches corresponds to 7£. Obtain this
point also by measurement, by laying off BF equal to CJE, pro-
longing EF to Gr and measuring GF. A C minus one-half GE
will equal the required distance. Repeat with several points
below E, and compare with the computed position of the lowest
point.
28. CRANK MOTION.
Apparatus. A steel scale AB, Fig. 24, divided into millime-
tres, slides -in a groove so that its position maybe read by an index
E. It is connected by the rod AD to the arm of the protractor,
whose centre is C. On turning CD, which carries a vernier F,
AB moves backwards and forwards. Several holes are cut in AD
so that its length may be altered at will.
Experiment. Make AD as long as possible. Measure CD by
turning it until D is in line with C and A, and read E\ then turn it
180°, and read again. One-half the
difference of these readings equals
CD. Next, to find the reading of
the vernier when CD and DA are
in line. Make ACD about 90°
and read E and F. Turn CD un-
til the reading of E is again the
same and read F. The mean of these two readings gives the re-
quired point. Repeat two or three times, and take the mean.
Let AS represent the piston rod of an engine, and CD the
crank attached to the fly-wheel. The problem is to determine the
relative positions of these two, during one revolution. Bring D
in line with CA, and move it 10° at a time through one revolu-
tion, reading E in each case. Do the same, using a shorter con-
necting rod, so that AD shall be about two or three times CD.
To compare these results with theory, first suppose the rod CD
infinitely long. The distance of AB from the mean position will
then always equal CD X cos A CD. This is readily computed
from the accompanying table of natural cosines. If, as is most
convenient, CD is made just equal to 1 decimetre, the distances are
given directly in the second column of the table by moving the
HOOK S UNIVERSAL JOINT.
69
Angle.
Cosine.
0°
1.000
10°
.985
20°
.937
30°
.866
40°
.766
50°
.643
60°
.500
70°
.342
80°
.174
90°
.000
decimal point two places to the right. Compare these results
with your observations. Construct a curve in which abscissas
represent' the computed positions of AJB, and or-
dinates the difference between the observed and
computed results, enlarging the scale ten times.
If a smooth curve is thus obtained it is probably
due to the short length of AD. The correction
due to this is readily proved to be AD —
*/AD2 — CD2 sin2 A CD, or calling the ratio
AD -^-CD—n, it is AD (n — J n* — mtf'ACD).
Compute this correction for every 30°, knowing
that sin2 30°= .25, sin2 60° = .75. The points
thus obtained should lie on the residual curve found above. Do
the same with the shorter arm AD.
29. HOOK'S UNIVERSAL JOINT.
Apparatus. A model of this joint with graduated circles at-
tached to its axles. The latter should be so connected that they
may be set at any angle.
Experiment. Set the axes at an angle of 45°, and bringing one
index to 0°, the reading of the other will be the same. Now move
the first 5° at a time to 180°, and read the other in each position.
Record the results in columns, giving in the first the reading of
one index, in the second that of the other, and in the third their
difference, which will be sometimes positive, and sometimes nega-
tive. Construct a curve with abscissas taken from, the first col-
umn, and ordinates from the third, enlarging the latter ten times.
It shows how much one wheel gets behind, or in advance of, the
other. To compare this result with theory, let Fig 25 represent a
plan of the joint, AC and CD being the two
axes. Describe a sphere with their intersec-
tion C as a centre. The great circle CD is
the path described by the ends of one hook,
CE that described by the other. D and E
must, by the construction of the apparatus,
always be 90° apart. Then in the spherical
triangle CDUwe have given J)E — 90°, ECD = 45°, the angle
between the axes, and one side as CD, and we wish to compute
.25.
70 COEFFICIENT OF FRICTION.
CE. But by spherical trigonometry, tang CE = tang CD cos
ECD. Substituting in turn CD = 5°, 10°, 15°, 20°, &c., we
compute the corresponding angle through which the second wheel
has been turned. Construct a second curve on the same sheet as
the other, using the same scale. Their agreement proves the cor-
rectness of both.
Experiments like Nos. 28 and 29 may be multiplied almost in-
definitely. Thus various forms of parallel motion, the conversion
of rotary into rectilinear motion by cams, link motion, gearing,
and, in fact, almost all mechanical devices for altering the path of
a moving body may be tested and compared with theory.
30. COEFFICIENT OF FRICTION.
Apparatus. AJ3, Fig. 26, is a board along which a block C is
drawn by a cord passing over a pulley J9, and stretched by weights
placed in the scale pan E. The friction is produced between the
surfaces of C and A J?, which should be made so that they may be
covered with thin layers of various substances as different kinds
of wood, iron, brass, glass, leather, <fcc. C is made of such a shape
that by turning it over the area of the surface in contact may be
altered. The pressure on C and the tension of the cord may also
be varied at will, by weights.
Experiment. Weights are added to E in regular order, as when
weighing, and the tension in each case compared with the friction
of C. Friction may be of two
kinds ; first, that required to start a
body at rest, called the friction of
repose, and secondly, the friction of
motion, or that produced when the
bodies are moving. To measure
the friction of repose, see if the
Fig- 26. weight is capable of starting the
body when at rest, if so, stop it and repeat, varying the weight
until a tension is obtained sufficient sometimes to start it and
sometimes not. This friction is very irregular, varying with dif-
ferent parts of every surface, and with the time during which the
two substances have been in contact. It is but little used practi-
cally, since the least jar converts it into the friction of motion.
The latter is much less than the friction of repose, and more uni-
k T
B
ANGLE OF FRICTION. 71
form. It is found by tapping the body so that it will move, and
seeing if the velocity increases or diminishes. In the first case
the weight in J^is too large, in the second too small.
The first law of friction is that the friction is proportional to
the pressure. The ratio of these two quantities is called the co-
efficient of friction. Make -the load on (7, including its own
weight, equal to 1, 2, 3, 4, 5 kgs. in turn, and measure the friction.
The latter equals the weight of E plus the load added to it minus
the friction of the pulley. If great accuracy is required, a table
should be prepared, giving the magnitude of the latter for differ-
ent loads. Compute the coefficient of friction from the observa-
tions, and if the law is correct they should all give the same re-
sult. Measure, in each case, the friction of repose and of motion,
and notice that the latter is always much the smaller.
Secondly, the friction is independent of the extent of the sur-
faces in contact. This law follows from the preceding, but it is
well to prove it independently by turning C on its different sides,
so as to vary the areas in contact. The friction will be found to
be the same in each case. Finally, measure the coefficients in a
number of cases, and compare the results with those given in the
tables of friction.
31. ANGLE OF FRICTION.
Apparatus. In Fig. 27, AB is a stand with an upright B C.
AD is a board hinged at A, which may be set at any angle by a
cord passing over the pulley C. The hinge is best made of soft
leather held by a strip of brass, and its distance from the upright
should be just one metre. A scale of millimetres is attached to
the upright, and a wire parallel to AD serves as an index. Evi-
dently the reading of the scale gives the natural tangent of the
angle of inclination DAB. A cord attached to D passes over
the pulley 6", around the wheel F, and is stretched by the counter-
poise 6r. F may be clamped in any position, or turned by a crank
attached to it. AD may thus be set at any angle, and its position
is to be determined when so inclined that any given body, as E, is
just on the point of sliding. E may be made exactly like the
sliding mass in Experiment 30, but to measure its friction of mo-
tion a fine wire should be attached to it and wound around the
axle of F. When the crank is turned raising AD, the body E is
thus drawn slowly down the inclined plane. In order that it may
not move too rapidly this portion of the axle should be much
72
BREAKING WEIGHT.
smaller than that around which the cord OF is wound. EF
should be a wire, as if a thread is used it will stretch, giving E an
irregular motion, alternately starting and stopping.
Experiment. To measure the coefficient of friction of repose,
turn the crank until the body begins to slide ; the reading of the
^ scale gives the tangent of
the angle of inclination.
But decomposing the weight
of E into two parts, paral-
lel and perpendicular to the
plane, their ratio will equal
the coefficient of friction,
and also the tangent of the
inclination. Hence the co-
efficient of friction is given
directly by the scale. Meas-
ure again the coefficients found in Experiment 30, and see if the
results agree with those then obtained.
32. BREAKING WEIGHT.
Apparatus. In Fig. 28, _Z? is a thumb screw, by which a spring
balance A may be drawn back so as to exert a strain on the wire
CD. Near C is placed a spring buffer so that when the wire
breaks, the jar may be diminished. D may be a simple peg to
which the wire is attached, or a spool with a clamp by which it is
held at any point. A convenient method of connection is to at-
tach one end of the wire to a chain, either link of which may be
passed over the peg J9, according to the length employed. To
test the accuracy of the balance a cord may be substituted for
CD, passed over a pulley E, and stretched by weights. Some
cord and fine copper or iron wire of various sizes should be fur-
nished, also a gauge to measure its diameter.
Experiment. Fasten a cord to (7, and pulling it over E, record
the reading of A, when by attaching weights, the strain is made
in turn 0, 5, 10, 15, 20, &c., pounds. To eliminate the friction of
the pulley, turn 7? first in one direction and then in the other, and
take the mean of the readings of A in each case. Now construct
a residual curve, in which abscissas represent the reading of A,
2) Jg
LAWS OF TENSION. 73
and ordinates the difference between this reading and the weight
applied. From this curve we can readily determine the true strain,
knowing the reading of A, how-
ever inaccurate the latter may be.
To measure the breaking weight
of any body, attach one end of it .
to C and the other to one link of u
the chain. Pass the latter over the
peg D, so that C shall be a short
distance from its spring buffer. Turn B and watch the index of
-4, until the cord breaks. A small block of wood may be placed
in front of the index to show the greatest tension attained, but
care must be taken that it is not disturbed by the recoil. Repeat
several times with other portions of the same cord and take the
mean of the observed maximum tensions. Do the same with some
specimens of wire, and compute their tenacity T, or strength per
square inch. For this purpose measure their diameter d with ac-
curacy, by the gauge, and calling TFthe breaking weight, we have,
33. LAWS OP TENSION.
Apparatus. In Fig. 29, A is a cast-iron bracket firmly fastened
to the wall. A hook is attached at_Z?, and from it the scale pan E
is hung by the wire or rod to be tested. Weights may then be
applied so as to give any desired tension to the latter. A brass
rod JBD hangs by the side of BC, being fastened to it by a small
clamp at the top. Fine lines are drawn on both rods, and their
relative change in position measures the elongation of J3C. Gr is
a reading microscope made of a brass tube about 6" long, with the
Microscopical Society's screw cut in one end, so that any micro-
scope objective may be used with it. A positive eyepiece with a
scale at its focus is slipped into the other end. This microscope is
mounted like a cathetometer, by fastening it to a vertical brass
tube screwed into the base of a music stand. It may be raised or
lowered, and held at any point by a set screw. The following
additional apparatus is also needed. Several wires or rods of va-
rious materials, as wood, copper, brass, iron, lead, and some of the
same material but different diameters. A millimetre scale to meas-
ure their lengths, and a Brown & Sharpe's sheet metal gauge to
give their diameters. Also a set of large weights to vary the ten-
74
LAWS OF TENSION.
a
Fig. 29.
sion. To prevent too sudden a jar on the wire, a board should be
placed under E to support it when a weight is added, then lower-
ing it by means of a screw.
Experiment. To measure the extension in any case, attach the
wire to be tried to the. hooks at IB and (7, and clamp JBD to its
upper end. Draw a line on J3 C opposite
one of those on J3D, and focus the micro-
scope G- on them. Read their relative posi
tion by means of the scale in 6r, then apply
the weight and read again. The length of
J3C is thus increased, while that of J3J) is
unaltered; hence the change in their relative
distances equals the extension. To reduce
this to millimetres, focus the microscope on
a standard millimetre, and thus measure the
scale in G directly. The distance from F to
the clamp is measured by the millimetre scale,
and the diameter of J3C\)j the gauge.
The laws of tension may now be determined.
1st. The extension is proportional to the length. Use a copper
wire about a millimetre in diameter, and mark on it a number of
lines at different heights. Measure the extension for each with a
load of 20 kgs. It will be found proportional to the distances
from the clamp. 2d. The extension is proportional to the weight
applied. Measure the deflection for the lower lines, increasing the
weight 2 kg. at a time, from 0 to 20 kilogrammes. Removing the
latter weight, see if the wire has returned to its original length ;
any increase is called the permanent set. See if the results agree
with the law. 3d. The elongations are inversely proportional to
the cross-section. Try the series of wires of the same material,
measuring the diameter of each with the gauge, and using the
same weight for all. The product of the square of the diameters
by the elongation should be constant. The modulus of elasticity
is the force which would be required to double the length of a
body of cross-section unity, supposing this could be done without
breaking it, or changing the law which holds for small weights.
To compute it, suppose d the diameter, I the length, and e the elon.
gation of a wire under a tension T. If the cross-section was
CHANGE OF VOLUME BY TENSION. 75
unity, to produce the same elongation we should increase the force
^T
in the same ratio as the two sections, or make it —- ; hence we have
4:T 4:1 T
e : I = — w : M* or M = — 75 . Measure this modulus for the
-a* Ttea *
various substances provided, and compare the results with those
given in the books. Finally, with an undue load the wire will take
a permanent set, which increases if the wire is stretched for a con-
siderable time. Study the laws regulating this property in the
case of lead, in which the set is very marked.
34. CHANGE OP VOLUME BY TENSION.
Apparatus. A rubber tube AB, Fig. 30, about a metre long,
and two or three centimetres in diameter, is closed above and be-
low by plugs. The upper one is perforated, and carries a glass tube
with graduated scale attached. A scale is also placed by the side
of the rubber tube, and a number of points are marked on the lat-
ter. A cord and friction pin C (like that of a violin) is fastened to
the lower plug, by which the tension of the tube may be varied.
On the other side of the scale is a square rod DE of elastic rub-
ber, about the size of the tube, and similarly marked. A pair of
outside calipers capable of measuring objects as large as the rod to
within a tenth of a millimetre, is also needed.
Experiment. The tube should be calibrated by weighing it
when empty, and when filled with water to the zero, or beginning
of the glass tube, also when filled to some division w, near its top.
Call these three weights to', to", wrrr. Then w" — w' — the weight
of a cylinder of water just filling the tube, or xr% in which Us
the known length, and r the radius. From the equation w" — wr
— Trr2/, we obtain r = t/ -- j - . The volume per unit of length
w" — wf . wm — w"
is -- 7 - , and in the same way for the glass tube it is -
Call the ratio of these two, or —77 - r' — = b. So much of the
w — w n
work may be done once for all. Any change in volume of the
interior of the tube can be accurately measured by noting the
change of level in the glass tube.
Fill the tube with water to the point marked n, and read the
position of the marks. Stretch it by turning the pin below so as
76
CHANGE OF VOLUME BY TENSION.
to lengthen the tube. Read each mark in its new position to-
gether with the water level, and so proceed, taking a number of
readings under different tensions.
Construct a curve in which abscissas represent
readings of the water level, and ordinates the
changes which take place in the length of the
rubber, draw also other curves, in which abscissas
represent the water level, and ordinates the
change of lengths of each section of the tube.
To do this it is most convenient to make a table
giving the readings of each point, a second
3 giving the difference of reading of each two con-
Fig, so. secutive marks, and a third giving the increase
of length they undergo when the cord is stretched. The first
of the curves will be nearly a straight line, and the tangent
of the angle it makes with the horizontal line, or the ratio of its
vertical to its horizontal progression multiplied by #, gives the ratio
of the increase of volume compared with the increase of length.
In the same way by the other curves, determine the change in
length of each section of the tube compared with the whole change.
Next measure the height of each marked point of the rubber
rod, also its diameter at these points. Stretch it and measure
again, and take four series of observations in this way. Now con-
struct curves, in which abscissas represent scale readings, and or-
dinates alterations in thickness as given by the gauge. The scale
for the latter must be greatly enlarged. Measure the area en-
closed by this space, and reduce it to square millimetres, allowing
for the change of scale. Multiplying this area by four times the
thickness gives approximately the diminution in volume due to the
contraction in the centre. If the rod is much altered in form,
the change in cross-section may be obtained more accurately by
taking the difference of the squares of the thickness before and
after extension. Using them as ordinates of the curve the vol-
ume is given by the enclosed area. Construct such a curve for
each extended position of the rod, and compare the decrease of
volume thus found with the increase due to the change of length,
or the product of the cross-section by the change of reading of
the lower index mark.
DEFLECTION OF BEAMS. 77
35. DEFLECTION OF BEAMS. I.
Apparatus. In Fig. 31, AB is a rectangular bar of steel rest-
ing on two knife-edges, with a load applied to its centre by a
weight placed in the scale-pan D. To measure the flexure, a mi-
crometer screw C is placed over the bar, and turned until its point
touches the latter. JZisa galvanic battery having one pole con-
nected with the bar, the other with C through an electro-magnet
F.. When the screw touches the bar the circuit is completed, and
the magnet draws down its armature with a click. This gives a
very accurate test of the exact position of the screw when contact
takes place. The length of the beam may be altered by changing
the position of A and J5. C and D are also movable, and a set
of weights is provided to vary the deflection. To ensure contact
the wire should be soldered in position, and the point of C tipped
with a piece of sheet platinum. A convenient size of bar for lab-
oratory purposes is about half a metre long, a centimetre wide,
and half a centimetre thick, using weights from 100 to 2000
grammes. Instead of the electro-magnet F, a galvanometer may
be used if preferred.
Experiment. Set A and IB 50 cm. apart, and C and D midway
between them. Turn C until it touches the bar, when instantly a
current will pass from the battery
E through the magnet F, making
a click, or if a galvanometer is used,
swinging the needle to the right or
left. Read the position of the mi-
crometer screw, taking the whole
number of turns from the index on
one side, and the fraction from the
graduated circle. Add 1000 gram-
mes to jf>, and bring the screw again
in contact. The difference in the
readings gives the deflection with great accuracy. The general
formula for the deflection a of a beam of length I, breadth #, and
TP78
depth <#, under a load TFJ is a — TCTS > in which D is the modu-
lus of transverse elasticity, or the weight required to make a
equal one, when b, d and £, all equal unity. From this formula
the following laws may be deduced. 1st. The deflection, when
small, is proportional to the weight applied. Measure the deflec-
78
DEFLECTION OF BEAMS.
tion for every hundred grammes, from zero to two kilogrammes,
and see if the beam returns to its original position when the load
is removed. If not, the change is the permanent set. Construct
a curve with abscissas proportional to the load, and ordinates to
the corresponding deflection. Evidently, according to the law,
this should be a straight line, and the near agreement proves con-
clusively its correctness. 2d. The deflection is proportional to
the cube of the length. Measure the deflection with a load of
2 kgs., changing the length of beam 5 cm. at a time, from 50 cm.
to 0, keeping O and D always at the middle of the beam. Care
must be taken in each case to first measure the micrometer-read-
ing when no load is applied, as this point will vary, owing to irreg-
ularities in the bar or stand. To compare the results with theory,
construct a curve in which abscissas represent lengths of the beam,
and ordinates deflections. According to the law this should be a
cubic parabola, having the equation y = axs. To find the value
of a, suppose one of the earlier readings gave a deflection of
5.8 mm., for a length, of 40 cm. ; then 5.8 = a 403, or a = .00009.
Substituting this value in the equation, construct the curve y =
.00009 x* on the same sheet with the experimental curve, and see if
they agree. To find the value of J), draw a line nearly coincident
with the first series of observations. Deduce from it the increase
of deflection for each added kilogramme. Substitute this value for
a in the formula, making TF— 1, and giving £, b and d, their proper
values, found by measuring the bar. D is now the only unknown
quantity, and may be obtained by solving the equation.
If desired, bars of different materials may be provided, and the
modulus of each determined by measuring the deflection with a
given load, and substituting these values in the formula. The law
that the deflection is inversely proportional to the breadth, may be
proved with bars alike except in breadth, and the law of the thick-
ness in a similar manner. The form of the beam when bent is found
by shifting the micrometer G and measuring the deflection at vari-
ous points between A and JB, and the effect of a change in position
of the load by moving D. The same apparatus may be applied to
the case of a beam supported at more than two points, or to beams
built in at one or both ends. This experiment may also be almost
indefinitely extended by using circular and triangular bars, hollow
DEFLECTION OF BEAMS.* 79
or solid, also those of a T or I shape, or, in fact, girders of any
form.
36. DEFLECTION OF BEAMS. II.
Apparatus. AB, Fig. 32, is the bar to be tested, which may be
one of the square rods described in the next experiment. It is
clamped at one end by placing it between two similar rods, one of
which is nailed to the wall, and the other pressed down upon it by
two or three clamps, like those of a quilting frame, as at O. A
small mirror maybe placed upon it at any point, and the deflection
measured by reading in it the reflection of a scale F by the tel-
escope E. The beam is bent by weights placed in the scale-pan D.
By attaching a finely divided scale to A, the absolute deflection
may be read by the telescope E, using it like a cathetometer.
Experiment. Clamp AB so that its length shall be 50 cm., and
place the mirror at its end. Place the telescope E opposite A and
focus it, so that on looking
through it the image of
the scale F shall be dis-
tinctly seen reflected in
the mirror. Read the po-
sition of the cross-hair in
the telescope to tenths of
a division of the scale.
Now place a kilogramme
in D. The beam is at once bent, and although the mirror moves
but little, the scale-reading is greatly altered. The work described
under the last experiment may be repeated with this apparatus.
Or to vary it, let the following measurements be made. Place 2
kgs. in J>, and take the scale-readings when placed at distances of
5, 10, 15, 20, &c., centimetres from B. Again, take the scale-read-
ing very carefully when no load is applied. Now place in D as
great a weight as the beam will safely bear, and take readings
every half minute for five or ten minutes. Then remove the load
and see if the beam returns to its original position. The perma-
nent set of the beam may be studied in this manner.
To compare the results with theory the scale-readings must be
reduced to angular deviations by the formula given on page 24.
For small deflections, however, it is sufficient to divide the change
80 TRUSSES.
in scale-reading by twice the distance EA^ to obtain the tangent
of the angle through which the mirror moves. In deducing
the form of an elastic beam by Analytical Mechanics, we have
El -7-3 = P(l — #), the moment of the deflecting force P. In
this E is the modulus of elasticity, I the moment of inertia of the
cross-section, y the deflection of any point at a distance x from
the end, and I the length of the beam. Integrating, we obtain
dil JP x2 dii
dx = ~EI^X — "2")' *n wn*cn d£" mav ke compared directly with
the measurement.
37. TRUSSES.
Apparatus. A number of deal rods, as nearly alike as possible,
half an inch square and five or six feet long. These are to form
the beams or units of which all the trusses are to be composed.
They may be connected by clamps like those used for quilting
frames, by boring holes in them and fastening by wire, or better
still by using small carriage:bolts, \" in diameter. A scale-pan and
set of weights serve to apply a strain not exceeding one or two
hundred pounds to any part of the truss to be tested. By attach-
ing a fine scale the deflection of any point may be read by a tel-
escope mounted like a cathetometer. The strain on any portion
may be determined by inserting a small spring balance, as will
be described below.
Experiment. This apparatus may be procured at very small ex-
pense, while with it almost all the laws of elasticity may be
proved, and the strength of a great variety of trusses for bridges
and roofs tested. Although the following work resembles that of
Experiment 35, yet its importance, and the different method of
measurement employed, justifies its repetition.
The flexure of a beam is proportional to the load. Set two knife-
edges 40 inches apart, cut a rod a little longer than this, and lay it
on them. Attach a fine scale to its centre point, focus the tel-
escope on it and record the reading. Add weights, a pound at a
time, until the beam breaks. The increase of reading in each case
over that given at first, is proportional to the load. It must be
noticed, however, that when the beam is much bent a new law
holds. Repeat these measurements with rods 30, 20, and 1 0 inches
TRUSSES. 81
long. The flexure is proportional to the cube, the breaking weight
inversely as the square of the length.
In the same way the effect of applying a load at different points,
or the deflection at different points with a given load, may be
measured. A long beam may also be supported at several points,
and the effect of a moving load noted. To measure the strength
of a beam built in at one or both ends, clamp it at those points
between two similar beams, one above, the other below, and fasten
them by bolts or clamps to a fixed upright. Having thus fully
determined the strength of the single beams, they should be ex-
amined when combined. Join two beams together so as to form a
T, and measure the strain necessary to pull the vertical one off.
Different kinds of joints may thus be compared. Some form of truss
should now be built, and its strength
tested. Let the king-post, Fig. 33, be
the form selected, and make the
span AS 40 inches. Cut two rods
a little longer than this, and bore the
holes A, B and D. Cut two more ,
rods CD with, holes distant 10
inches, and attach them to the rig ^
others by bolts at D. Add AO
and CB, and connect the two trusses thus formed by cross pieces
at A, B, C and Z>, so that they shall be ten inches apart. A
small bridge is thus made whose stiffness is remarkably great.
Such a structure should bear a weight of fifty, or even a hundred
pounds, with little flexure. Measure the deflection at different
points under varying loads, trying the effect of applying the latter
at the centre, on one side, or distributing it uniformly.
The force acting on any beam is readily determined by replacing
it by a spring balance, with a screw and nut which may be made
to take up the whole strain, without distorting the structure.
If the force is one producing compression, it is best to elongate the
beams, as at (ZEJ and insert the balance between C and JS. Com-
pare the various results with those obtained by computation. It
must be remembered that this method of connecting the beams of
a truss by bolts is not employed in actual practice, but it is very
convenient, and sufficiently strong for a model. If preferred,
6
82 LAWS OF TORSIOX.
proper tools may be supplied, and the student may frame his truss
as in a real bridge or roof. Further, any beam, as CD, subjected
only to tension, may be replaced by a wire.
The rods will also be found useful for a great variety of pur-
poses. Thus one of the easiest ways to make large screens is to
fasten four of them together at the ends, and attach thick paper to
them by double-pointed -tacks. Again, a convenient way to make
a large rectangular box to cut off light, or to protect an instru-
ment from dust, is to connect twelve of these rods at the ends by
slipping them into corner pieces of tin, Fig.
34, and covering them with black paper.
In the same way all the principles of
framing may be taught, and quite compli-
cated structures built. It is well to have
some of the latter loaded until they break,
to determine the weakest points. They are
then easily repaired by inserting new pieces
of wood. Another very good object to construct and test is a sus-
pension bridge. Use two stout wires or chains for the' suspension
cables, and build the roadway of the rods, hanging it from the
chains by wires with screw threads cut on their ends, so that their
lengths may be adjusted by nuts. Test the strain on the chains
by inserting a spring balance like that known as the German ice-
balance, and measure the deflections of the different parts under
varying loads.
38. LAWS OF TORSION.
Apparatus. Let AB, Fig. 35, represent the bar whose torsion
is to be measured. The farther end B is firmly fastened to a
piece of wood (7, which can turn around the axis of the beam, but
may be clamped in any position to the semicircle immediately be-
hind it. A in the same way is attached to J9, which carries two
brass rods, acting like the dog of a lathe. EF is a long rod
mounted on an axle, which may be turned by placing weights in
the scale-pan Q. The latter is supported by a cord passing over a
curved block at the end of E, so that the moment of the weight,
or its tendency to twist the bar is unchanged, whatever the posi-
tion of EF. To measure the angle of torsion, two mirrors are
attached to AJB, and the scales H,Hf reflected in them are viewed
by telescopes T, I'. By making H, H! arcs of circles, with centres
at A and B, the angle of torsion may be obtained directly.
LAWS OF TORSION.
83
Fig. 35.
Experiment. To measure the torsion of the beam AB, attach
it to the stand, as in the figure, placing the mirrors at a distance
apart equal to the length
to be examined. Focus
the telescopes I, I' so
that the scales H, Hf
shall be distinctly visible,
and read the position of
the cross -hairs. Place a
weight in the scale-pan
6r which will twist both
the mirrors A and J?, de-
viating the former the
most. See how much
each scale-reading has
changed ; their difference
measures the angular
twist of AS. This may be reduced to degrees from the radius
of curvature of the scale, and the magnitude of its divisions, recol-
lecting that the motion of the mirror is only one-half that of the
reflected ray. From the theory of torsion the following laws are
readily deduced. The angle of torsion is proportional to the mo-
ment of the deflecting force. To prove this law, measure the tor-
sion with several different weights in 6r, and see if the angles are
proportional to the weight. The distance of the point of applica-
tion of Gr from the axis may also be varied, and it will be found
that the torsion is proportional to the product of this distance mul-
tiplied by 6r. The torsion is proportional to the length of the bar.
Prove this by varying the distance between the mirrors, leaving
the bar unchanged. By using a variety of bars it may be proved
that in those having similar sections the torsion is proportional to
the fourth power of the similar dimensions. In rectangular bars of
breadth £, and depth c?, it is proportional to -^bd (b2 + (#2), and in
tubes to ^7r(>4 — /4) calling r and / the outer and inner radii. In
practice, owing to the warping of the surfaces, these formulae un-
dergo slight modifications.
04: FALLING BODIES.
39. FALLING BODIES.
Apparatus. This consists of two parts, a chronograph capable
of measuring very minute intervals of time, as hundredths of a
second, and the arrangement represented in Fig. 36, for making
and breaking an electric circuit when the body falls. A ball A is
attached to the spring D by a short thread and wire. Burning
the thread the ball is released, and the spring rising allows the cur-
rent to pass from the battery B, to (7, Z>, E, F, and the chronograph
G. This marks the beginning of the time to be recorded. Its
end is shown by the breaking of the circuit, which occurs at F
when A strikes E. To have the current broken during this time,
instead of closed, it is merely necessary to place the points E and
F below the springs. Another method of releasing the ball is to
put an iron pin in its upper side, and support it by an electro-
magnet. The instant the current is broken it will fall. One of
the best forms of chronograph for this purpose is that devised by
Hipp, in which a reed making a thousand vibrations a second re-
places the pendulum of an ordinary clock. It therefore ticks a
thousand times a second, and measures small intervals of time with
great precision. Very good results may be obtained with a com-
mon marine clock, removing the hairspring and replacing it by an
elastic bar of steel. An electro-magnet is placed close to the pallet,
so that when the current passes the spring is bent. The clock,
therefore, starts the instant the circuit is broken, and stops as soon
as it is closed.
Experiment. The time of falling of any body through a height
equals y— , in which g = 9.80 metres. Measure the time of
Y y
fall through various heights by altering
the length of the wire AD, noting the
height in each case with care, repeating
several times and taking the mean. Com-
pare this with the result given by theory.
In vacuo the time would be independent
of the material or magnitude of the ball.
In air, however, this is not the case, owing
to the resistance. The latter may there-
fore ke determined by measuring the time
of fall of bodies having the same form but different weights. This
apparatus may also be applied to the measurement of the velocity
of curve-motion and personal equation, or the coefficient of friction
METRONOME PENDULUM. 85
of a body may be found with great accuracy by measuring its time
of descent down an inclined plane.
40. METRONOME PENDULUM.
Apparatus. A light deal rod is provided, to which two
leaden weights may be attached at any point by set screws. A
knife-edge passes through the centre of the rod, so that it may be
swung like a pendulum. Either weight may also be attached to
the knife-edge by a fine wire. A millimetre scale is used to meas-
ure the distance of the weights from the knife-edge, and a bracket
fastened to the wall and carrying a steel plate serves as a support.
Experiments. First, to prove the law of the single pendu-
lum. Attach the heavier weight to the knife-edge by the wire,
using as great a length as possible. Measure the time in sec-
onds of making 100 single, or 50 double vibrations, also the dis-
tance from the centre of the lead to the knife-edge. Repeat with
several lengths of wire.
Now compare the observed time with that given by the formula
jl
t = TT t/-, in which g = 9.80 m., and I is the measured length.
T t/
Repeat one of these measurements with the lighter weight, using
the same value of I. It should give the same result as the other.
Next place both weights on the deal rod at opposite ends. Meas-
ure, as before, their time of vibration, also their distances from the
knife-edge. Compute the time of vibration as above, merely sub-
20 7' 2 I 'w"lffz
stituting for I the value — /,/ , — rnrri in which «/, w" are the
W i ~\~ W I/
weights, and Z', I" their distances from the knife-edge. If greater
accuracy is required, a third term must be introduced for the
weight of the rod. Repeat with various positions of the two
weights, and compare the results with theory.
41. BORDA'S PENDULUM.
Apparatus. In Fig. 37, CD is the pendulum, formed by attach-
ing a ball of lead C, to a wire nearly four feet long, and supporting
it on a knife-edge D. A sheet of platinum is fastened below the
ball, so that when at rest it dips edgewise into a mercury cup,
making electrical connection with the battery B. E is a clock
whose pendulum F dips in a second mercury cup. When both
pendulums are at rest the current passes from B through (7, J), E
86
BORDA S PENDULUM.
and 2>\ to 6r, which is an electric bell arranged so as to strike
whenever the circuit is closed.
Experiment. Connect the battery IB with the wires attached to
C and G-. The bell will instantly strike. Start the pendulum
D C. Whenever it passes through the mercury cup, that is, with
every swing, the electric current passes through G and makes the
bell strike. Stop CD and start the clock. The strokes now occur
at intervals of exactly one second. Now set both pendulums go-
ing. The bell will strike only when
both are vertical at the same instant.
This will occur at regular intervals,
equal to the time required by the
longer pendulum to lose just one
fijyjuN vibration. Record the minute and
(JUT \ second of each stroke for ten or fif-
teen minutes consecutively. The
first differences give the intervals,
and from the mean of the latter the
time of vibration may be computed
with great accuracy. For exam-
ple, if the interval is 47 seconds it
denotes that in this time CD made one less, or 46 vibrations, hence
the time of a single vibration would be f$=1.0217 seconds. An
error of one second in the mean of the interval would make the
time £f=1.0213 seconds, or alter the result less than a two-thou-
sandth of a second. The method, therefore, is one of extreme pre-
cision. Sometimes, especially when the pendulum is swinging
through 4 small arc, the bell will strike for several consecutive sec-
onds, owing to the considerable interval of time during which con-
tact is made at C", so that for several seconds the circuit is closed
at F before it is broken at C. In this case the time of the first
stroke should be recorded and their number ; the true time being
taken as the mean of the first and last. To make CD vibrate in
one plane instead of describing an ellipse, attach a fine thread to
the ball (7; draw it to one side about ten inches; let it come to
rest, and then burn the thread. Finally measure the length I of the
pendulum, or the distance from the knife-edge to the centre of the
Fig. 37.
i/
T
TORSION PENDULUM. 87
ball, and compute the force of gravity g from the formula ; t =
—, in which t equals the time of vibration, and TT — 3.1416.
y
This experiment may be repeated with a different length of pen-
dulum, or it may be varied so as to prove that the time increases
with the amplitude. In the latter case the arc through which the
ball swings should be as large as possible, and it should be meas-
ured as it progressively diminishes. To compute the theoretical
time of swinging through any arc a, divide versin \ a, or the ver-
tical distance through which the ball moves, by its length, and call
the quotient x. Then the time if for any value of x may be found
from the equation if = (I + | x + ITS & + &c.) £, in which t is
the time when the arc is very small. When a = 180°, or the ball
swings through a semicircle, t' = 1.180 £, when a = 30°, if =
1.0063 t, when a = 10°, if = 1.00067 t, hence for small arcs the
correction for this cause is very small. If great accuracy is re-
quired in this experiment the suspending wire should be very
light, and with the knife-edge should vibrate in about one second
when the ball is removed, or a correction may be applied for them
as described in Experiment 40.
42. TORSION PENDULUM.
Apparatus. AB, Fig. 38, is a vertical wire with an index C,
which moves over a graduated circle. Weights of a cylindrical
form, as .Z>, may be attached below in such a manner that the
wire cannot twist without turning them. To vary the length of the
wire it is passed around several small brass tubes E, F, G-, placed
at different heights, so that it may be clamped at these points by
inserting a pin G- passing into a hole bored behind them. A scale
and clock beating seconds are also needed for this experiment.
Experiment. 1st. The time is indepen-
dent of the amplitude. of the vibration. Use
the whole length of the pendulum, and apply
such a weight that the time of a single vi-
bration shall be about one second. Twist
the index through a small arc, and take the
time of one hundred oscillations by noting
the position of the index at the beginning of
a minute, and the exact time, when after
making one hundred single oscillations, it
88
TORSION PENDULUM.
again reaches the same point. Dividing the interval by one hun-
dred gives the time of a single oscillation. Repeat two or three
times with arcs of different magnitudes, and compare the results.
2d. The time is proportionate to the length of the wire. Make
the same experiment, first with the wire of its full length, then,
passing the pin through the different tubes E, F, clamping it at
these points. Measure their distances from J5, and compare with
the law. In the same way the relation of the time to the diameter
of the weight, or to its length, may be tested and compared with
theory.
MECHANICS OP LIQUIDS AND GASES.
43. PRINCIPLE OP ARCHIMEDES.
Apparatus. An inverted receiver J., Fig. 39, with a stopcock,
or better, an J" gas valve below. Near the top is placed a hook
C with a sharp point, which is used to mark the level of the liquid.
The whole may be hung from the scale-pan D of a large balance,
EF, which has a counterpoise attached to the other end. G is a
beaker to collect the water drawn off, and H a stand by which A
may be supported if necessary. A set of weights is needed, also
two bodies M and JVJ one heavier, the other lighter than water.
They may be made of metal and wood, or, if preferred, of glass,
and loaded so that one shall float, the other sink.
Experiment. 1st. A heavy body when immersed is buoyed up
by a force equal to the weight of the displaced liquid. Place the
receiver on the stand, fill it with water and
draw out the latter until the point of the hook
just touches the surface, observing the point
of contact, as in Experiment 13. Place the
beaker Gr on the scale-pan J>, suspend M be-
low it, and add weights to the other side so
as to bring the beam into equilibrium. If
now the receiver is brought up under M the
water will rise, and the equilibrium will be
destroyed. Open £ and draw off the water
into 6r until M, being immersed, the level is
again exactly at C. Now replacing Gr on D
it will be found that the equilibrium is re-
stored. Hence the loss of weight of JJf equals K ^
the gain of 6r, or the weight of the displaced
liquid, since the level is unchanged.
2d. Since action and reaction are equal, the vessel appears to
90 RELATION OF WEIGHTS AND MEASURES.
gain in weight by an amount just equal to the loss of M. Sus-
pend A from the scale-pan, and M from the stand. Bring the
water-level to C and counterpoise as before. Immerse M, when
the water will rise, and the weight apparently increase. Open B
therefore, and draw out the water until the level is restored, when
it will be found that the beam is again balanced, showing that it
was necessary to draw out a volume of liquid equal to that of M.
3d. A floating body displaces a weight of liquid just equal to^
its own. Rest A on its stand and restore the water level to C.
Place G and JV" on the scale-pan and counterpoise. Let N float in
A, open IB until the proper level is attained, collect the water in
G, and replacing the latter on the scale it will be found that the
equilibrium is restored. That is, the weight of the displaced wa-
ter, or the increase of G, equals the weight of N".
44. RELATION OF WEIGHTS AND MEASURES.
Apparatus. A delicate balance with a counterpoise on one side
and scale-pan on the other, below which a small cube of brass is
suspended by a very fine platinum wire. In addition, a beaker
containing distilled water, a thermometer and a set of weights,
must be provided.
Experiment. By definition a gramme is the weight in vacuo
of a cubic centimetre of distilled water, at the temperature of
maximum density, that is, about 4° C. ; the object of the present
experiment is to test this relation. Add weights to the scale-pan
until equilibrium is established ; then immerse the cube in the dis-
tilled water, first washing it with caustic potash to remove the air,
then very thoroughly with common water to remove the potash,
and finally with distilled water. The weight now required to
counterpoise it will be greater than that previously taken, by an
amount equal to the weight of the displaced water. Record the
height of the barometer and the temperature of the water. Next,
to determine the volume of the cube, measure the twelve edges
very carefully to tenths of a millimetre, and take the mean of each
set of four which are parallel. The product of these three means
equals the volume. The dividing engine should be used to attain
sutficient accuracy in this measurement. Add to this the volume
of the wire found by multiplying its cross section by the length
HYDROMETERS. 91
submerged. Correct the weight found above for the buoyancy of
the air, and the volume for the dilatation of the water, as in Ex-
periment 19. Only in this case the whole weight of the displaced
air must be added, since by definition the weight must be taken in
vacuo. The density I) of the water at any ordinary temperature
t, is given by the formula D = 1 — .000006 (t — 4)2, its density
at 4° being unity. After applying these two corrections, see if the
volume of the water in cm8 equals its weight in grammes.
By using English weights and measures instead of French, the
relation between the inch and pound may be established in a
similar manner.
45. HYDEOMETEES.
Apparatus. This consists of four tall jars, two containing
water, the third some light liquid, as alcohol, and the fourth a
saturated solution of salt, or other heavy liquid. A variety of
hydrometers, some giving the specific gravity directly, others with
the scales of Beaume, Cartier and Beck, &c. In one of the jars
of water, which should be larger than the other, is a Nicholson's
hydrometer, Fig. 40, and on the table a box of weights, a small
stone and a piece of hard wood. Near by should be a sink, with
a large jar in it, through which water is continually flowing, to
wash the hydrometers.
Experiment. Float each hydrometer in turn in the jar contain-
ing water, and record the reading of the point of the scale on its
stem just at the surface. This point is determined most accurately
by bringing the eye nearly on a level with the top of the water,
but a little below it. All should give a specific gravity of very
nearly unity, the difference being partly due to error in the instru-
ment, and partly to expansion of the water by heat. Next im-
merse each in the alcohol, take the reading and wash by plunging
it in the large vessel of water. Do the same with the solution of
salt. If any hydrometer sinks lower than the top of its scale, the
liquid is lighter than it can measure; if it floats too high the liquid
is too heavy. Finally, reduce all the readings to specific gravities
by the hydrometer tables. These instruments being of glass are
easily broken, and must be handled with care.
Turning now to the Nicholson's hydrometer, place weights on
the upper scale-pan A, until it sinks to the mark scratched on its
92
SPECIFIC GRAVITY BOTTLE.
stem. Record their sum, and replace them in their box, taking
care (as must always be done with delicate weights) never to
touch them with the fingers, but only with forceps. Moreover
they must never be laid down on the table, and to prevent their
falling into the water, the piece of metal C must be kept over the
mouth of the jar. Place the stone, or other object whose specific
gravity is to be measured, on A, and add weights, as before. Call
their sum in the first case w, in the second wf. Raise
the hydrometer out of the water (of course first re-
placing the weights in their box), and place the stone
on the lower scale-pan B. Immerse it, taking care
that there are no adhering air bubbles. If these can-
not be detached with the finger, remove the stone and
wash it first with caustic soda, and then with pure
water. Call w" the weight required to immerse the
hydrometer when the stone is on JB. Then w" — w'
is the apparent diminution of weight of the stone when immersed,
or the weight of an equal bulk of water. As w — wf is the weight
of the stone, its specific gravity is // — /. Perform the same
experiment with the piece of wood, only placing it below IB to
keep it down, and noticing that w" will be greater than w.
46. SPECIFIC GRAVITY BOTTLE.
Apparatus. A balance weighing up to 100 grms., and turning
with two or three milligrammes, a set of weights and a specific
gravity bottle, or as a substitute, two glass stoppered bottles, the
neck of one being large, of the other, small. They should be care-
fully selected, with stoppers fitting smoothly, and a scratch should
be made both on the neck and stopper, so that the latter may
always be turned into the same position. As objects for determ-
ination of specific gravity any liquid may be used, as a solution of
salt, and two or three solids, as stones, coins, gold ornaments,
sand, &c.
Experiment. Weigh the empty bottle and stopper, and call
their weight w&. Fill the bottle with water, insert the stopper and
wipe off the liquid which has overflowed, taking care that the ex-
terior of both bottle and stopper are perfectly dry. Call this
weight ww.
HYDROSTATIC BALANCE. 93
Fill with the liquid to be tested in the same way, taking care
that the stopper is inserted in the same position as before, and that
no liquid adheres to the exterior. Let the weight be wit then
w\ — wa and ww — w& are the weights of equal bulks of the liquid
w\ — ioa
and of water, and the specific gravity of the liquid is = w w •
To find the specific gravity of a solid, use the bottle with the
larger neck. Call w the weight of the solid, ww the weight of the
bottle filled with water, and we the weight when the solid is in-
serted, and the remaining space filled with liquid ; then w + w^ —
w6 equals the weight of a volume of water equal to that of the
solid, and the specific gravity = — -r— - — . This method is
J w -f- wv — io8
applicable to solids heavier or lighter than water. The principal
precaution is to take care that no bubbles adhere to the solid or
sides of the bottle, and that the stopper is always pressed in by the
same amount. Use the same devices for removing the air as with
the Nicholson's hydrometer, Experiment 45. With metals these
precautions are especially important, or large errors will be intro-
duced. Another good method is to place the flask containing the
solid and water under the receiver of an air-pump and exhaust
two or three times. This method is not applicable to wood, as it
removes the air from the cells, and increases the apparent specific
gravity. The same effect is produced by long immersion, and
finally when waterlogged, the specific gravity becomes greater than
unity, and the wood sinks.
47. HYDROSTATIC BALANCE.
Apparatus. A complete apparatus for this purpose, known as
Mohr's Balance, may be obtained, and the following description is
especially applicable to it. A common balance may, however, be
substituted, raising one scale-pan and attaching a hook below. In-
stead of riders it is then generally more convenient to use ordinary
weights. Some solids and liquids are also needed as substances
whose specific gravity is to be determined.
Experiment. Attach the small scale-pan to the left, and the
glass counterpoise to the right end of the beam. The weighing
is done by riders, of which there are three sizes, whose weights are
in the ratio 10, 100 and 1000. The beam is divided into 10 equal
parts, so that when balanced the weight may be read off directly
94 EFFLUX OF LIQUIDS.
to three places of decimals. Fill the small jar with water, and see
what weight is necessary to immerse the counterpoise. It will be
found to be very nearly 1000, and evidently equals the weight of
the water displaced. Next, fill the jar with the liquid to be tested,
and see what weights are now required. The ratio in the two
cases is the specific gravity. The temperature should be recorded
in each case by the thermometer contained in the counterpoise,
and if great accuracy is required a correction applied for it, or bet-
ter, the liquids may be cooled to. the standard temperature.
To find the specific gravity of a solid, wind a piece of fine
wire around it, and suspend from the left hand end of the beam.
Counterpoise by adding lead, sand or paper to the scale-pan at
the other end until a perfect balance is obtained. Immerse in a
vessel of water, and balance by adding the riders ; their weight
equals that of an equal volume of water. Then remove the solid,
and again bring the beam to a horizontal position by the riders ;
this gives the weight of the solid, which divided by the weight of
the water displaced, gives the specific gravity. If more convenient,
the weight of the body may be obtained directly by the riders with-
out counterpoising it.
Next, find the specific gravity of a piece of wood, or other solid
lighter than water. Attach a piece of lead, or other body heavy
enough to sink it, and measure, as above, the following quantities.
"Weight of solid in air t0s, weight of lead in air t01? weight of lead
in water w{, weight of solid and lead in water wlsf. Then w\ — w\
= weight of a bulk of water equal to that of the lead. wl + wa
— wia' — weight of a bulk of water equal to lead and solid. Hence
their difference, or wl -f- w6 — wla' — wl -f- w' = w& + w{ — wls' =
weight of water equal in bulk to solid, and weight of solid divided
by this, equals specific gravity, or $. G-. = — ^ — r r
The same precautions are necessary, as with the gauge flask,
regarding air bubbles, and the riders should never be touched with
the fingers, but always with a small bent wire.
48. EFFLUX OF LIQUIDS.
Apparatus. In Fig. 41, A and IB are two reservoirs of tin, or
wooden boxes lined with lead, each containing two or three cubic
EFFLUX OF LIQUIDS.
95
feet. "Water is admitted by a valve at (7, and passes through a
cylinder of perforated tin .Z>, to break up the stream and prevent
much motion of the water in A. An outlet is made at E^ which
may be closed by a stick of wood with a rubber flap on its end J£,
which is held in place by the pressure of the water. To keep the
level constant, a funnel F is connected with the interior by a rub-
ber tube, so that it may be raised or lowered, and serve as an over-
flow, or a simple straight tube may be used, passing through the
bottom of A to the surface. The height of the water is read by a
hook gauge Gf with an index attached, moving over a scale. A
number of brass plates fitting into .2? are provided with orifices of
various shapes and sizes, some circular, rectangular and triangular,
and others furnished with projecting cylindrical or conical tubes.
The second reservoir JS has also a hook gauge and scale H to
show the amount of water in it, and an outlet I closed by a plug.
To prevent motion of the surface of the water around H, a dia-
phragm is placed in the centre of the reservoir, on which the water
impinges, a number of holes being bored in the lower portion to
equalize the level on each side.
Experiment. When water flows through an aperture in a thin
plate the amount per minute is much less than that given by the-
ory, owing to the contraction
of the liquid vein immedi-
ately after leaving the orifice.
The ratio of the two is called
the coefficient of efflux, and
the whole science of hydrau-
lics is based on this constant.
To determine it, water is al-
lowed to flow from A under
a given head through an ori-
fice E, and the quantity meas-
ured by the scale attached
to H. Place one of the cir-
cular orifices in E, and meas-
ure its height by bringing the water just on a level with it, and
using the hook gauge. This is done as is described in Ex-
periment 13, by bringing the point of the hook just to the surface
of the liquid, so as slightly to distort the image of outside objects,
and reading the position by the scale. Close E with the rod I£
and open the valve (7, first raising the funnel F nearly to the top
Fig. 41.
96 EFFLUX OF LIQUIDS.
of the 'reservoir. When the water begins to escape over the edge
of the funnel close the cock, and read very carefully the level by
the gauge. Read also the height of the liquid in _B, which should
be nearly empty. At the beginning of a minute open E by re-
moving the rod J£, when the water will begin to flow into £ in a
clear transparent steam, marked, when the aperture is not circular,
by alternate swellings and contractions. As the liquid will at once
descend in A, the valve C should be opened at the same time, and
adjusted so that the water shall slowly trickle over the edge of the
funnel, or outlet tube, or the latter may be dispensed with, and the
surface kept just at the point of the hook. When IB is nearly full,
which should take at least five minutes, close E and note the time.
It is best to make this come at the end of a minute. Now read
the height of the water in _#, empty it, and repeat to see if the
same results are obtained twice in succession. Make the ex-
periment again with other pressures, also changing the orifices.
To reduce the scale-readings of H to cubic inches, the reservoir
3 must next be calibrated. If nearly rectangular, a direct meas-
urement will give its horizontal cross-section, but if the sides are
at all curved it is safer to use some other method. A plan much
used in practice is to mount it on a platform scale and weigh it
when empty, and when filled with water to various heights, and
reduce the weight of the water in each case to cubic inches, by
dividing by .03614, the weight in pounds of one cubic inch of wa-
ter. A curve should then be constructed, in which ordinates rep-
resent the scale-readings and abscissas the volumes. If greater
accuracy is required, the tenth of a cubic foot used in Experiment
19 should be employed. A T is placed between its valve and the
glass, the branch of which is connected with the hydrant by a
rubber tube. It is then hung over the reservoir J5, as in the fig-
ure. To use it, admit water until it is filled to the top of the hook
in its upper end. Shut off the water, and open the valve below.
When the water level has reached the lower point, close the valve
and read the gauge in J?, thus taking a series of readings which will
correspond to intervals of precisely one tenth of a cubic foot. In
this case it is best to construct a residual curve to show more
clearly the irregularities in form of the reservoir.
The area of the orifices must next be measured with a fine scale,
JETS OF WATER. 97
reading to tenths of a division by the eye, or if greater accuracy
is required, using the dividing engine.
Finally, to compute the theoretical flow, we have the following
data. By the theorem of Torricelli the velocity = JZgh, in
which g — 32.2 ft., or the acceleration of gravity, and h is the
height of the liquid above the centre of pressure of the orifice. This
equals the difference in the two readings of the hook gauge in A,
before and after the experiment, correcting for the positio.n of the
centre of pressure, which will sensibly coincide with the centre of
gravity of the orifice. Thus with a circular orifice • one-half its
diameter must be subtracted. A stream of water will then flow
out having a volume equal to that of a prism with cross-section s
equal to that of the orifice, and a length v for each second, or in
t seconds, the observed time, the volume V should be stv =
st+/%gh. The observed volume is obtained directly from the cali-
bration of 7?, of which either a curve or a table should be fur-
nished. This quantity divided by "Fgives m, the coefficient of efflux.
49. JETS OP WATER.
Apparatus. A cylindrical brass tube is used as an orifice, and
is mounted at a height of three or four feet from the floor, with a
hinge and graduated circle, so that it can be set at any given
angle. A deal rod divided into inches is attached to it to measure
the range, and the whole is connected with the hydrant by a rub-
ber tube and valve, so that water may flow through it at any re-
quired velocity. The water is collected as it escapes in a large
vessel, which is weighed in a spring balance before and after the
experiment, and thus the amount of water determined. A second
scale of inches is also required to measure the vertical descent of
the curve.
Experiment. Almost all the laws of projectiles may be proved
by this apparatus. 1st. The form of the jet is a parabola. Set the
tube horizontal, and allow the water to flow through it, with such
a velocity that in moving three feet horizontally it will descend
about the same distance. Take care that this velocity is un-
changed during the experiment by noticing that the horizontal
range remains the same. Now measure the vertical fall of the jet
for every two inches on the horizontal scale, and construct a curve
with these distances as coordinates. Next, to measure the veloc-
7
98 RESISTANCE OF PIPES.
ity, allow the water to flow into the vessel for one minute, and
weigh it. The weight in grammes equals the number of cubic
centimetres, and this divided by the area of the orifice (found by
measuring the diameter of the tube), gives the velocity of the
water per minute. Divide this by 60, for the velocity per second,
and construct the parabola given by theory, in which x = vt, and
at* gx2
y = -Q- = Q-g, and the acceleration of gravity g = 386 inches.
Great care must be taken to reduce all these quantities to the same
measure, as- inches or metres, several different units being pur-
posely employed in these measurements. Repeat the latter part
of this experiment with three or four different velocities, and see
if for a given value of y, x is proportional to v.
2d. The horizontal range for a velocity v, and angle of projec-
v2
tion «, equals — sin 2a. Prove this by measuring the range for
y
every 5° from 0° to 90°. Evidently the maximum is when x =
45°. In a similar manner we may prove that the maximum range
on an inclined plane is attained when the direction of the jet
bisects the angle between it and the vertical, and again, that the
curve of safety or envelope' to all the parabolas formed with a
given velocity when the jet is turned in different directions, is a
parabola, with the orifice for a focus.
50. RESISTANCE OF PIPES.
Apparatus. A f" brass tube six feet in length has five holes
drilled in it at intervals of exactly a foot, taking care that no burr
or roughness remains on the inside. Short pieces of brass tubing
are soldered on over them, and long glass tubes are attached by
pieces of rubber hose. The whole is mounted on a stand, so
that the brass pipe is horizontal, and the glass tubes vertical and
a foot apart. Each tube is graduated, or has a paper scale at-
tached, .to show the height at which the water stands in it. Wa-
ter may be passed through the brass pipe at different velocities by
connecting it with the hydrant, and regulating the flow by the fau-
cet. To keep the pressure regular, it is better to connect with a
separate reservoir, and to measure the velocity, the water may be
received in a large graduated vessel.
Experiment. When water flows through the brass pipe it will
rise in the glass tubes owing to the friction, and the latter may be
FLOW OF LIQUIDS THROUGH SMALL ORIFICES. 99
very accurately measured by the height of the liquid. On trying
the experiment it will be noticed that the top of the liquid col-
umns lie very nearly in a straight line, passing through the open
end of the pipe, where of course the pressure is zero. The exact
pressure should be measured by the attached scale, and observa-
tions of all of them taken for several different heights. A second
series of experiments should also be made to determine the veloc-
ity corresponding to these heights. In this case the escaping
liquid is received in the graduated vessel for a known time, or the
time required to fill it is noted, and from this, knowing the volume
and cross-section of the pipe, the velocity is readily determined.
The results should be represented by curves, first making abscissas
distances, and ordinates pressures, and secondly, using velocities
as abscissas, and the heights of the liquid in the most distant tube
for ordinates. From these curves the laws and coefficients of
liquid friction are readily determined. i
51. FLOW OF LIQUIDS THROUGH SMALL ORIFICES.
Apparatus. A Mariotte's flask is placed about three feet above
the table and a rubber tube is connected with its outlet. To this
is fastened a brass tube with a perforated screw cap, so arranged
that small circles of platinum foil may be inserted, with holes of
various sizes. A vertical scale shows the height of the orifice, and
a balance serves to measure the quantity of water received.
Experiment. Fill the Mariotte's flask with water. For this
purpose it is often convenient to have a third tube, which is closed
by a rubber cap, except when the flask is to be filled. It is then
opened to allow the air to escape, and water is admitted by one of
the other tubes. Raise the orifice so that water is just on the
point of flowing out of it, and measure its height. Insert one of
the platinum diaphragms and lower it, so that the water shall flow
out drop by drop. Collect what escapes during a minute, and
weigh it. Lower the orifice and repeat at intervals, until it is as
low as possible. Measure also at the point where the drops begin
to unite into a continuous stream. For all lower points measure
the length of the stream, that is, the distance before it begins to
divide into drops.
Compute the coefficient of efflux by means of the usual formula,
100 CAPILLARITY.
Y
V = mstv = mst*/2gh, hence m = .~ », in which s equals the
7T6?2
cross section = ^T, calling d the diameter of the orifice, t — the
time of flow = 60, h the head, or the reading first taken minus
that corresponding to the given observation, and V is obtained
from the weight, remembering that 1 gramme of water = 1 cm3 =
.061 inches. Finally, construct a curve in which ordinates rep-
resent the coefficients m, and abscissas the heads h.
By this simple apparatus, interesting results could be obtained by
measuring the flow of various liquids with different pressures and
orifices. Their relative viscosity might thus be compared.
52. CAPILLARITY.
Apparatus. In Fig. 42, A is a tall bell-glass set in a glass jar
IB containing water. C is a glass tube drawn out to a point and
Connected with A by a rubber tube ; it is immersed in a test tube
J>, containing the liquid to be tried. A may be filled with air by
blowing through the bent tube E. Paper scales divided into milli-
metres are attached to J3 and D to measure the pressure, and D is
supported in such a way that it may be raised or lowered at will.
Experiment. Draw out a piece of glass tubing to a fine point,
break off a small piece and grind the end flat so that the orifice
shall be circular and smooth. Connect
it as at (7, by a rubber tube, with the
bell-glass A^ and fill the latter with air by
blowing into E. Raise the test-tube D
containing the liquid to be employed, so
that the air escaping from C shall bubble
up through it. Soon the pressure in A is
so far diminished that it becomes insuffi-
cient to overcome the resistance opposed to it, the flow will then
stop, and the top of the liquid in C w;ll be found to be very much
curved. Record the pressure of the air in A which equals the
difference in level of the water within and without it. Call it A,
and call hf the difference in level within and without C. Repeat
this observation several times, either by blowing into E, or by
lowering D so that the flow shall recommence. Next remove the
tube from the liquid, break off the end, and stick it carefully into
PLATEAU'S EXPEKIMENT. 101
a cork. Grind down the end of C until it is again flat, and repeat
until observations have been obtained with orifices of five or six
different sizes. Now place the cork on the dividing engine, Ex-
periment 21, and measure the diameter of each of the ground ends.
This may be obtained with great accuracy by placing the axis of
the tube vertical so as to look down through it.
Let s be the specific gravity of the liquid and x the height to
which it would tend to rise in the tube, if the bore were the same
throughout as at the end. The pressure due to this force will
then be sx, and in the same way the pressure due to the column
hr will be sh'. Both of these pressures will be in equilibrium with
the force h of the water in JB. In other words, h = sx -\- shf, or
x _ _ L_? froni which x may be calculated in the various cases.
s
If the liquid in D is water, s = 1, and x = h — hf. This method
of studying capillarity was first proposed by M. Simon, who, how-
ever, found that his results did not agree with those obtained by
direct measurement. It has, however, the great advantage that
the diameter may be obtained with accuracy, even with very mi-
nute tubes, and the latter being heated to redness are rendered
chemically clean.
53. PLATEAU'S EXPERIMENT.
Apparatus. Some of Plateau's soap-bubble mixture, formed by
mixing pure oleate of soda with 30 parts of water, and adding two
thirds its bulk of glycerine. The oleate is made of olive oil and
soda, which is then filtered. Common soap may however be used.
Wires are bent into the following forms and soldered at the cor-
ners. A tetrahedron with a single wire as a handle, a, cube, a cir-
cle, two triangles hinged along one side, and two squares, made in
the same way, also a small vertical stand arranged so that two
circles may be placed on it at any height. To measure the figures
obtained, an upright (7, Fig. 43, is attached to the table to support
a sheet of paper and at a distance of about two feet is a second
support, A, in which is a small hole to look through. A third
stand, B, serves to hold the wire figures in any desired position.
Experiment. Dip the tetrahedron into the liquid, and on draw-
ing it out, films will be found extending from each of the six
edges, and meeting in the centre. This point is a fourth of the
distance from each face to the opposite angle. Attach the tetrahe-
102 PLATEAU'S EXPERIMENT.
dron to B, so that one face shall be nearly horizontal, and one
edge perpendicular to the line through ABC. On looking
through A it is projected as a triangle on (7. Move _Z?, if nec-
essary, so that its lower face shall be projected as a straight line.
Attach a piece of paper to C
and mark on it the corners of the
tetrahedron, also the intersection of
the films. Measure on the paper the
distance of this point from the top
and bottom of the figure. Their
ratio should be one to three. Turn the tetrahedron around, and
repeat the measurement with one of the other sides. It should
be the same for all.
A general law of these films is that they are always subjected to
tension and continually tend to contract, owing to the molecular
attraction of the particles. This may be shown in various ways.
Attach a loop of the finest silk thread to the circle of wire. Dip it
in the liquid, and a film will be obtained in which the loop will float,
irregular in shape and in any position. Break the film inside the
loop, and instantly by the contraction of the film around it, it will
be drawn out into a perfect circle, leaving of course a hole in the
centre. Inclining the circle from side to side the loop moves
freely over the film, presenting the curious appearance of a sheet
of liquid containing a moveable hole.
Immerse the tetrahedron again in the liquid. The six films pul-
ling equally in 'opposite directions, hold the centre point in equi-
librium. Now break one of the films, and the remainder con-
tracts, forming a curious curved surface drawn towards one side
by a single plane film. On breaking this second film, the surfaces
again contract and form the warped surface known as the hyper-
bolic paraboloid.
Immersing the cube in the same way twelve plane surfaces are
obtained, meeting in a small square in the centre. This square
may be parallel to either face, aud may be made to alter its posi-
tion by gently blowing, so as in appearance to split it. See how
many different figures can be obtained by breaking one or more
films, and draw them in your note book. The whole number is
twelve, not including a single plane attached to one face only. If
PNEUMATICS. 103
the films attached to two opposite parallel sides are broken, a plane
is obtained supported between two curved surfaces, the intersec-
tions being curved lines. Draw these lines by attaching the cube
to B and see if they are hyperbolas. Another curious effect is ob-
tained by blowing a small bubble and attaching it to the centre
square, when it assumes a cubical form with curved sides ; in the
same way a four-sided bubble may be formed with the tetrahe-
dron. Similar figures may be obtained with an octahedron,
or other figures, but they are more complex.
On dipping the two triangles into the liquid a film forms over
both, and on increasing the angle between them a single plane film
is found attached to their common side, which is split as they sepa-
rate. Breaking this film the curve springs back as before, forming
a very beautiful hyperbolic paraboloid. This is probably the best
way of producing this warped surface, and its properties are well
shown by it. Varying the angle between the triangles, its form, or,
more strictly, its parameter may be altered at will. Make the
angle between the triangles about 30°, and draw the curve of in-
tersection of the plane film with the other, also a section through
the centre at right angles to it. Try and determine the form of
the first of these curves, and see if it is a circle, parabola or hyper-
bola. Now break the film and draw the enveloping curves on the
same sheet as before, to show how the films have contracted.
Do the same with the jointed squares. Place the two circles
on their stand near together, blow a bubble and lay it on them.
Then draw them apart, and a hyperboloid of revolution of one
nappe will be obtained.
54. PNEUMATICS.
Apparatus. The object of this experiment is to familiarize the
student with the ordinary lecture-room apparatus in pneumatics,
and is therefore chiefly of value to those who propose to adopt
teaching as a profession. The apparatus needed will depend on
the objects of each student, but may be made to include almost all
the instruments used in a full course of lectures on this branch of
physics. The following description, however, applies only to such
experiments as could properly be introduced in any common
school. The most important instrument is of course the air pump,
which need not be of large size, or (for most of these experiments)
capable of producing a very high degree of exhaustion. The
104 PNEUMATICS.
other apparatus needed is best determined from the following list
of experiments, which may be varied almost indefinitely.
Experiment. Place a receiver on the pump-plate, taking care
that no dust or grit is retained under the edge, which should be
freely supplied with sperm oil, or tallow, to ensure contact. Open
communication between the pump and receiver, and close that
leading to the outer air. Exhaust, by working the handle of the
pump, and see if any leakage takes place around the bottom of the
receiver, in which case air bubbles will be seen forcing their way
through the oil. The greatest trouble in using the air-pump is to
make this joint tight, especially if the plate or receiver is not
ground perfectly true. When the exhaustion is nearly complete
the pump handle will work freely, until the very end of the stroke,
when a slight hissing will be heard, due to the expulsion of the
remaining air. For this reason the piston must be moved until it
strikes the end of the cylinder each time, and the strokes must be
taken steadily, and not too fast. When the air is removed the
exterior pressure becomes so great that it is impossible to move the
receiver without breaking the glass. On opening communication
with the outer air, the latter rushes in, and the receiver is easily
removed. To determine the degree of exhaustion, a syphon
vacuum gauge may be employed. This consists of a bent glass tube
like a syphon barometer, with the closed end only about half a
foot in length, and containing mercury, which of course rises to the
top. Place it under a receiver and exhaust, when it will be found
that as soon as the pressure inside is reduced to less than six inches
the mercury begins to fall, until in a perfect vacuum it would stand
at the same height in both branches of the tube. Read the differ-
ence in level, which in a common pump should not exceed two or
three millimetres. If a barometer gauge, or long tube dipping in
mercury, is attached to the pump, subtract its reading from that of
the standard barometer, and the difference should equal that of the
syphon gauge. Place a beaker of water on the pump-plate with a
bolt head (or tube with a bulb blown at one end) in it, cover with
a receiver and exhaust slowly. The air will now bubble up
through the water, owing to its tendency to expand when the
outer pressure is removed. If the pump is a very nice one, this
PNEUMATICS. 105
experiment, and others requiring water, should be omitted, as the
vapor may rust the interior of the pump. On readmitting the air
the water will rush up into the bolt-head until but a small bubble
of air remains. The ratio of the volume of this bubble to the
whole interior of the bolt-head, shows the degree of exhaustion.
When nearly all the pressure of the air is removed from the sur-
face, the water bubbles make their appearance in it, due to the
dissolved air. Carrying the exhaustion still farther, vapor begins
to be formed so rapidly that the water enters into ebullition. This
effect is more easily obtained if the water is somewhat warm.
Select two tubes about three feet long, and closed at one end, fill
one, j5, with mercury (Experiment 58), the other, ^4, with air, and
dip both into a small vessel containing mercury. Cover them with
a tall receiver and exhaust. The mercury will descend in 13 until
nearly on a level with that in the cistern, the air meanwhile escap-
ing from A in bubbles. Readmit the air and the mercury will rise
in both tubes, that in A being the lowest. Any leakage in the
pump is well shown in these experiments, as it will cause the li-
quid to begin to rise slowly as soon as the pumping stops. To see
if the leak is in the pump, or under the receiver, close the connec-
tion between them when leaks in the latter only will be percepti-
ble. The great pressure of the air may be shown in various ways.
Thus the palm-glass is a cylindrical vessel open at both ends,
which is placed on the pump-plate and closed above by the hand ;
after exhaustion the latter is removed only with difficulty. Re-
placing the hand by a sheet of rubber, a single stroke of the pump
will draw it strongly inwards, and in the same way a tightly
stretched bladder may be made to burst with a loud report. In
the upward pressure apparatus the air, being withdrawn above a
piston, the latter, with a heavy weight attached, is raised by the
pressure of the air below. The Magdeburg hemispheres consist
of two brass hemispheres, accurately ground together, which re-
quire a great force to separate them when the air is with drawn
from the interior. Great care is needed in handling this apparatus
as a slight blow will bend the brass sufficiently to cause leakage.
Bursting squares are sealed rectangular vessels of glass, which
explode when placed under an exhausted receiver. To prevent
injury they should be covered with wire gauze, and the orifice
106 PNEUMATICS.
leading to the purnp, protected by a brass cap and valve. The
porosity of wood may be shown by the mercury funnel, in which
mercury is driven lengthwise through a piece of wood which
passes through the top of a ' receiver. If a piece of wood held
under water is covered with a receiver, and the air exhausted, tor-
rents of bubbles imprisoned in its pores will pour from it. Now
on admitting the air the water enters the wood, which becomes
water-logged, and no longer floats. The revolving jet is a bent
brass tube, like a Barker's Mill, which when placed under a receiver
turns rapidly in one direction when the air is exhausted, and in
the other when it is readmitted. The effect of the resistance of
the air is shown by two fan wheels with vanes set flatwise and
edgewise, respectively. If set in motion, the former stops first in
air, but both revolve for nearly the same time in a vacuum. The
same effect may be shown by a feather and guinea placed in
a long glass tube from which the air is removed. They then
fall with nearly equal velocity from end to end. An import-
ant experiment is the proof of the weight of the air. A glass
sphere is weighed when full of air and when exhausted, and the
difference gives approximately the required weight. The exact
weight is obtained only by an accurate correction for temperature,
pressure and moisture. The two following experiments, though
properly belonging to other branches of physics, are inserted here
for convenience. Both require a very high degree of exhaustion.
If a bell is rung in a vacuum, no sound is heard. An electric bell
is most convenient for this experiment. It should be carefully
supported, so that the sound shall not be transmitted directly to
the pump plate. For this purpose it is sometimes hung by threads ;
a rubber support is also recommended. The experiment is gen-
erally more successful, if after exhaustion hydrogen gas is ad-
mitted, and the exhaustion repeated. It is, however, almost im-
possible to destroy all sound. The latent heat of aqueous vapor is
well shown by the experiment of freezing water in vacuo. A
shallow pan of concentrated sulphuric acid l is placed on the pump
plate, and on this a wire triangle which supports a flat metallic
1 In all cases where sulphuric acid is used to absorb moisture in the presence of metallic
surfaces, it should be freed from nitric fumes by boiling it for some time with sulphate
of ammonia.
107
dish holding the water to be frozen. The whole is covered with a
small receiver, and exhausted quickly. On removing the pressure
from its surface the water is rapidly converted into vapor, which is
absorbed by the sulphuric acid as fast as formed ; the action there-
fore continues, the latent heat being obtained at the expense of the
water, which accordingly cools until it is converted into ice. By
substituting for water more volatile substances, as a mixture of
solid carbonic acid and ether, and adding protoxide of nitrogen, the
most intense cold yet observed is attained. In the best pumps
water may be frozen by its own evaporation, without employing
acid to absorb its vapor.
55. MAKIOTTE'S LAW.
Apparatus. A modification of Regnault's apparatus may be
made chiefly with steam fittings, as shown in Fig. 44. C is a tall
mercury gauge formed of glass tubes, connected together by a
steam pipe coupling with red sealing-wax. If very high, all the
joints must be made like those of Regnault's gauge, but this is
unnecessary for pressures below one hundred pounds. A and B
are two similar tubes about three feet long, closed above by "pet-
cocks," and attached below by " unions," so that they may be easily
removed. E is a reservoir made of 3 inch pipe with caps, to hold
the mercury, and with an £" valve below, so that it may be
emptied if necessary. It is filled by removing the plug in the T
at G. I is a small force-pump such as is used in testing gauges,
by which water may be drawn from the reservoir K, and forced
into E. The water is allowed to flow back by opening the valve
H. Remove the plug 6r, and pour into E enough mercury to fill
A, £ and C. Work the pump slowly until E is full of water.
Then close 6r and expel any air that may remain by working I
and opening H alternately, until no air bubbles rise up through
the water in K. Scales are attached to A, IB and (7, and the first
two should be carefully calibrated (Experiment 10). IB may be
permanently filled with dry carbonic acid, or other gas.
Experiment. A must first be filled with dry air. For this pur-
pose connect" it above with a U-tube containing chloride of lime,
open the pet-cock and pump up the mercury nearly to the top,
thus forcing out the air. Open II a very little, and let the mer-
cury slowly descend. The air is thus drawn into A, first being
dried by passing over the lime. Repeat several times to expel all
the moisture that may remain, and finally, when full of dry air,
close the pet-cock. Read very carefully the height of the mer-
108
MARIOTTE'S LAW.
Fig. 44.
cury A, B and C, and record in three columns. Work the pump
a few times, and take readings at intervals of about 10 inches,
until the mercury has nearly
reached the top of A C. Note
the height of the barometer and
the readings of C and A, when
the latter is open to the atmos-
phere, also the height of the
mercury in A when standing at
the same level as in C. Write
in the 4th and 5th columns the
pressure in each case, found by
adding to the height of the ba-
rometer the difference in level
of the mercury, columns. In the
6th and 7th columns give the
volume of the gas in each case,
deduced from the table of calibration of A and B. Next take
the product of the pressure and volume, which would be constant
if Mariotte's law were correct, or the volumes inversely as the
pressures. Finally, construct curves for the two gases, making
abscissas represent these products, and ordinates pressures. These
results will be only approximate, owing to the change of tempera-
ture the gas undergoes when rarefied or condensed. To diminish
this error an interval should be allowed for the gas to attain the
temperature of the air of the room, or better, A and B should
be surrounded with a water jacket, the temperature carefully
noted, and a correction applied.
Much greater accuracy is attained by the following arrange-
ment. A third tube is employed, in the upper part of which two
platinum wires pointing upwards are inserted, the volume above
them being determined very accurately by inverting, and weigh-
ing the mercury required to fill it. This portion of the tube is
then enclosed in a larger one, through which water is kept circu-
lating, and its temperature noted by a thermometer. Fill the
tube in the usual way with dry gas, then condense it until the
mercury is just on a level with the upper platinum point. The
mercury in C should now stand near the top of the tube. Open
GAS-HOLDER.
109
IT, and let the pressure diminish until the mercury in the third
tube is exactly on a level with the lower platinum point. Record
the pressure in each case. To bring the mercury to the exact
level, raise it by the pump and lower by opening If until the
point is just perceptible by the slight distortion it produces in the
image of objects reflected in the surface of the mercury. Let the
pressure diminish to a few inches of mercury, let out a little gas
and repeat. The law may be tested for pressures less than one
atmosphere by merely drawing off the mercury in C until it stands
below that in the other tube. The ratio of the volumes being
constant in this experiment, the ratio of the pressures would be
its reciprocal, if Mariotte's law were correct. The deviation may
be shown by a curve in which abscissas represent the smaller
pressure, and ordinates the ratio of the two pressures.
56. GAS-HOLDEK.
Apparatus. A good gas-holder containing three, or better, five
cubic feet, with scale attached, the bell properly counterpoised,
and most important of all, the friction reduced to a minimum. To
calibrate it, the standard tenth of a cubic foot of Experiment 19 is
employed, and to measure the pressure some very delicate form of
gauge should be provided.
Experiment. The gas-holder consists of a large bell, AB, Fig.
45, suspended in a circular trough
of water, and counterpoised by the
weight F attached to a cord passing
over the pulley D. A curved piece of
metal E, called the cycloid, is attached
to this pulley, and carries a second
weight G-, which acts at a longer and
longer arm as the bell rises. It thus
compensates for the diminution of
weight of the bell when submerged?
nd renders the pressure nearly the
same whether the holder is full or
empty. The proper form for E is the
involute, a curve in which the perpen-
dicular on the tangent is proportional
to the angle described by the radius
Fig. 45.
110 GAS-HOLDER.
vector. The gas is drawn out by a large tube opening into the
bottom of the holder at j5, and covered at K by a cap with a
water seal. A large tube with a stopcock H, also opens' out of it,
through which the gas may be drawn, or if preferred, through
a small tube just below it. To fill the holder with air, remove IL
and press down on F, when the bell will rise to the top ; it may
be kept there by replacing J^ and closing H. If gas is to be
used, it must be filled through H, but this is a much slower process.
A variety of experiments may be performed with this apparatus.
First, test the holder. Fill the bell nearly full of air, depress
it a little by the hand, let it return, and record the reading of the
scale. Then raise it, let it descend, and again read. Repeat sev-
eral times, and the difference in the results shows the greatest
error due to friction. Do the same with other parts of the scale.
Next, calibrate the bell. The same method is employed as in Ex-
periment 48, only the air is collected instead of the water. In this
case, after emptying the holder, add one tenth of a cubic foot of air
at a time, and read the scale after each addition. Repeat drawing
out one tenth of a cubic foot from the holder into the glass stand-
ard, and see if the readings are the same as before. The great
difficulty in this experiment is the change of volume of the air
due to changes of temperature. As the bell rises from the water
the adhering moisture evaporates, and sometimes lowers its tem-
perature very rapidly. It is, however, customary to assume that
the air is saturated with moisture, and at the same temperature as
the water with which it is in contact.
Next, measure the pressure for different parts of the scale to see
if the compensation is exact. The gauge is attached to a small
tube just below If, with an independent outlet from the bell. To
save time the pressure may be observed after adding each tenth
of a foot in the last experiment. Various forms of gauges may be
employed. The simplest is a large U-tube, with a scale attached
to each branch. The pressure may thus be determined within a
hundredth of an inch. For greater accuracy, a bell glass standing
in water fnay be connected with the holder, and the difference in
level of the water within and without it gives the pressure. By
using two hook gauges for this purpose great accuracy may be
attained. A method in common use is in principle similar to the
GAS-METERS. Ill
wheel barometer. A small bell is connected with the interior of
the holder, and its rise and fall is measured by a cord passing over
a pulley which moves a pointer over a graduated circle. If the
pressure increases as the holder rises, the weight G should be
increased, and the contrary if it diminishes. The pressure to which
the gas is subjected is varied by changing the weight F. Prove
this, and determine the law. Do the same for different parts of
the scale. To test the above work, fill the holder with air, and
open J:Tvery slightly, or better, allow the air to escape through a
minute aperture. The holder will now slowly descend, and by
noting the time the index passes each tenth of a foot mark, a
series of numbers is obtained whose first differences would be con-
stant, if the apparatus was perfect. By varying the pressures, the
orifices and the kind of gas in the holder, all the laws of the flow
of gases may be verified.
57. GAS-METERS.
Apparatus. Two gas-meters, one wet and the other dry, both
graduated so as to read to thousandths of a foot. They are con-
nected together so that the gas will pursue the following course.
It leaves the pipe through a \fr valve, passes through a T to the
wet meter, thence through a second T to the dry meter, and by
a stopcock and third T to a fishtail burner. A short piece of pipe
is screwed into the open end of each T, which may be closed by a
cap, or connected with a gauge formed of a U-tube by a piece of
rubber tubing.
Experiment. Gas-meters are of two kinds, wet and dry. The
former consists of a cylindrical vessel half full of water, in which
is placed a rotary drum with four compartments. As these are
filled in turn, the drum revolves, and the amount of gas con-
sumed is measured by the number of revolutions. The dry meter
resembles in principle a blacksmith's bellows reversed in such a
way that air being forced into the nozzle, the handle moves up and
down. The number of strokes is then recorded by clockwork and
dials. The wet meter was first used, but is now superseded in
houses by the dry meter, owing to the error introduced by any
increase or diminution in the amount of water present. The former
is, however, still in vogue for experiments, as by it small amounts of
gas may be measured with much greater accuracy. To determine
112 GAS-METERS.
the amount of gas which has passed through the meter, subtract the
reading at the beginning from that at the end of the experiment,
or if the rate of flow is required, take readings at intervals of one
minute, as in Experiment 5. Usually in the best meters, one rev-
olution of the large hand equals one tenth of a cubic foot, and the
dial being divided into a hundred parts gives thousandths. In
meters intended to be used in houses, one revolution of the hands
of the three lower dials equals 100,000, 10,000 and 1000 cubic feet,
respectively, and a fourth dial is placed above, whose hand
makes one revolution for every five cubic feet, and which is used
in testing the metre.
The common method of testing a meter is to bring the upper
hand to the zero, connect it with a4 gas-holder, and force air or gas
through it until the reading is the same as before. The change of
reading of the holder should now be just five feet, and the differ-
ence is the error of the meter. This experiment should be re-
peated two or three times. If the meter reads to thousandths of
a foot an additional test is needed to see if the divisions of the
large dial correspond to equal quantities of gas. For this purpose,
allow the gas to flow very slowly through both meters, turn it off
and read them, dividing the thousandths into tenths by the eye.
Allow a few thousandths to pass and read again, and so take a
series of readings, until two complete revolutions have been made.
Represent the results by a residual curve, in which abscissas repre-
sent the readings of the large hand of the wet meter, and ordi-
nates the difference between the two enlarged, moving the origin
down so as to bring the points on the paper. Two curves are thus
obtained, one for each revolution, which should be coincident ex-
cept for the accidental errors, and their deviation from a straight
line shows the inequality of the thousandths, as given by the dry
meter.
The next thing to be determined is the loss of pressure due to
each meter. Evidently a certain amount of power is necessary to
overcome the friction, and this power is obtained at the expense
of the pressure of the gas, which therefore leaves the meter with
less pressure than it enters it. To measure this, connect the first
and second T with the two arms of the gauge, and allow the gas
to pass through the meter. The difference in level of the two
GAS-METERS. 113
tubes shows the loss due to the meter. See if this varies with dif-
ferent pressures and with different positions of the revolving drum.
By using the second and third T, the dry meter may be tested in
the same way.
When a metre is placed between the outlet and the valve by
which the gas is turned off, an error is introduced whenever the
latter is opened or closed. This is due to the difference of pressure
within and without the revolving drum, produced by gas flowing
in or out without in some cases moving the hand. To show this,
close both the valve and the stopcock near the third T. Open the
valve, the gas will rush into both meters, moving the hands a
small amount. Close the valve and open the stopcock, when the
gas will rush out until the pressure within the meter equals that
of the outer air. Take a number of readings, opening them thus
alternately. Each meter is here affected by the error caused by
the other, and by the intermediate pipe, to eliminate which the
valve and stopcock should be placed close to the meter to be
tested. The error may then be determined for different pres-
sures and different positions of the hand. This error is not cu-
mulative, and seldom exceeds one or two thousandths of a foot.
To measure the amount of gas consumed by any burner under
different pressures, connect it with one of the meters, and attach
the gauge to the third T. Turn on the gas and take a five minute
observation ; that is, take six consecutive readings of the meter at
intervals of one minute, also the pressure as given by the gauge.
Do the same with several other pressures, and see if the flow is
proportional to the square root of the pressure, or if the curve
formed by the readings of the gauge and volumes of gas burnt is
a parabola. A similar experiment may be performed with an
aperture in a plate of platinum, and the height of the flame meas-
ured corresponding to different pressures and rates of consumption.
A regulator of some form, such as will be described under the
photometer, should be introduced to prevent accidental variations
in the pressure, if great accuracy is expected in the last experi-
ment. The laws of the efflux of gases may then be tested, or the
uniform division of the meter, by allowing the gas to escape very
slowly, and seeing if the volume is proportional to the time.
8
114 BAROMETER.
58. BAROMETER.
Apparatus. Two barometer tubes, one already filled and
placed in its cistern, some pure mercury, and a stand by which the
height of the mercury column may be measured. This stand may
be made in a variety of ways. Thus a half metre steel bar divided
into millimetres is fastened to an upright, and a slider attached to
it, so that it may be set at any desired height. This slider carries,
first a steel point about 40 centimetres below, to determine the
height of the mercury in the cistern ; secondly, a vernier or a brass
plate with a single line cut on it and resting against the steel
scale, and finally two index plates of brass between which the tube
is placed. The slider is raised or lowered until a thin line of light
is just visible between the top of the mercury and the bottom of
the index plates, and the reading then taken by the vernier. The
tubes are held in position either by rings of brass, or by strips
fastened by hinges. A steel rod three or four decimetres long and
a tall jar of water, are also needed.
Experiment. First, to find the distance from the steel point to
the index plates. This may be done by the cathetometer, or by
the second steel rod. Place the jar of water close to the upright^
and bring the points of both rod and slider just in contact with
the surface of the liquid. Read the vernier, lower the slider until
the index plates are just on a level with the top of the steel rod,
and read again. The difference added to the length of the rod
equals the distance from the index plates to the lower point. The
length of the steel rod is found by bringing the index plate first to
its upper and then to its lower end.
Now place the filled barometer in its proper position between
the index plates, move the slider down until the point just touches
its reflection in the mercury, and read the vernier. Raise the
slider, until placing the eye on a level with the index plates the
light is just cut off between them and the mercury. The differ-
ence between these readings is the height through which the
slider has been raised, and this added to the distance from the
plates to the point gives the height of the mercury column. Now
measure the height of the standard barometer placed with the
other meteorological instruments. Reduce it to millimetres (1 me-
tre = 39.37 inches), and the difference of the two is the error of
the barometer first measured. It is probably due to a little air in
the top of the tube.
BAROMETER. 115
Now fill the empty tube in the following manner. If the
mercury is not perfectly pure, it must be cleaned as described in
Experiment 9. Hold the tube with the left hand in an inclined
position, the closed end resting on the table. Pour in mercury
slowly to within a few inches of the top. To prevent spilling, the
stream should be guided by the forefinger and thumb of the left
hand held at the opening of the tube. Next close this opening
with the finger and raise the closed end so that the bubble of air
shall move slowly along the tube. Make it pass from end to end,
until all the small adhering air-bubbles are removed. Then fill it
full of mercury, and closing the end again invert it, and immerse
in the cistern, removing the finger under the surface of the mer-
cury. The latter will now descend in the tube until its height is
about 30 inches, leaving a vacuum at the top. Its pressure at the
bottom of the tube is then just equal to that of the atmosphere,
which by pressing on the outside mercury, supports the inner col-
umn. The vacuum at the top is known from its discoverer as the
Torricellian vacuum, and is one of the most perfect that can be ob-
tained artificially. To see if any air has entered, incline the tube,
and notice if the mercury rises to the top, remaining at the same
level throughout, and if when made to oscillate gently it strikes
the top with a sharp click ; if not, air has entered, and the experi-
ment must be repeated. Next, put this tube in the place of that
previously filled, measure its height and determine the error.
Allow a bubble of air to enter one of the barometer tubes (the
one in which the error is the greatest), and notice that it increases
in volume as it rises until it reaches the top, when it causes a con-
siderable depression of the mercury column. Repeat, until this
has fallen eight or ten inches. Then measure the height with care,
and suppose it to be an observation taken at the top of a moun-
tain. Compute the height on this supposition by the method given
below. The temperature of the- air and mercury at the upper sta-
tion may be assumed equal to that of the thermometer outside the
window, and at the lower station to those obtained by direct ob-
servation. This work may well be supplemented by a determina-
tion of the altitude of a real mountain. For this purpose the
pressure of the air must be measured at the top and bottom, either
by an aneroid, or more accurately by a mercurial mountain barome-
116 MEASUREMENT OF HEIGHTS BY THE BAROMETER.
ter. When going on such an expedition, it is well to take also a
hypsometer, and other instruments, so as to determine the dew-
point, solar radiation, temperature of the air, etc. These will
be described in detail under Meteorological Instruments. On
reaching the foot of the mountain, observations should be taken,
and again on the return, and the mean of these compared
with those taken at the top. Or better, one observer with a ba-
rometer is left below to take readings at regular intervals, as every
quarter of an hour, during the whole time of the ascent. These
are afterwards compared with those taken at the same time at the
summit. Of course the lower barometer is compared carefully
with the others at the beginning and end of the trip, and the
errors corrected. If only one barometer is at hand, and time al-
lows, a series of observations should be taken before and after the
ascent, a curve constructed, and the intermediate readings ob-
tained by interpolation. Accuracy is to be expected only from a
long series of observations above and below, by which accidental
errors are eliminated ; any sudden change in the weather, as a
thunder-storm, is especially liable to affect the result.
A small aneroid which may be easily carried in the pocket, is
often very serviceable in preliminary surveys ; by using it in con-
nection with a pedometer, an approximate profile of the country
may be constructed. In the same way the variations in the grade'
of a railway may be determined. The delicacy of these barom-
eters is such that they will show the difference of the level of the
different parts of a house, or even the rise and fall of a vessel at
sea. For such observations the height is obtained with sufficient
accuracy by allowing 87 feet for every tenth of an inch fall of the
barometer.
MEASUREMENT OF HEIGHTS BY THE BAROMETER.
On ascending from the surface of the earth, the barometric pres-
sure continually diminishes, and this is due to the fact that being
caused by the weight of the superincumbent air, the greater the
height the less the load to be borne. The law of diminution is easily
deduced by the calculus ; call p the pressure in inches at any height
J3~. The decrease of pressure, or — dp, in any interval dJT, is evi-
MEASUREMENT OF HEIGHTS BY THE BAROMETER. 117
dently due to a column of air of this height, whose weight is pro-
portional to p. Hence — dp = apdff, dff= —*-t or H = - log p
+ O. The constant a equals the pressure due to a column of air
of height unity and under pressure unity, or its reciprocal equals
60,300. The elevation E, or difference in height of two points
J^and II' is therefore H — Hr = 60,300 (log/ — log p).
In order to obtain the true height by the formula, it is necessary
to apply several corrections, of which the most important are the
following.
I. Capillarity. The effect of this force is to depress the mer-
cury column by an amount dependent on the diameter of the tube.
A constant quantity should therefore be added to each reading.
Unfortunately this result is modified by the adhesion of the liquid
to the tube, which renders this correction uncertain ; sometimes,
therefore, the height of the meniscus or curved portion is allowed
for, but the best way is to use a very large tube, when the effect of
capillarity becomes inappreciable.
II. Temperature of the Mercury. The standard pressure as-
sumes the temperature to be 0° C. ; at higher temperatures the mer-
cury would be lighter, and the pressure less. Let p be the observed
height at temperature T, and P the true height with mercury at
*n
zero, then p = P (1 + a T), or P — + . — m, in which a equals
the expansion of the mercury per degree. As the scale expands
also, allowance must be made for this, which gives a = .00009,
when the scale is of brass. The temperature T is given by the
thermometer attached to the instrument, and this correction
should always be applied to a mercurial barometer, but not to an
aneroid, when the height is wanted.
III. Temperature of the Air. In the above discussion the air
also is supposed to be at zero. If warmer it will be lighter, and
the elevation greater than that here assumed. Call t and if the
temperatures above and below, and their mean t" — ^ (t -\- tf).
The true elevation E' will then equal E (1 -f~ a t") in which
a = ^3- the coefficient of expansion of air. This is the most im-
portant correction of all, and should always be applied, or large
errors will be introduced.
118 BUNSEN PUMP.
IV. Latitude. Still another correction may be applied when
great accuracy is required, owing to the diminution of the force
of gravity as we approach the equator. The computed elevation
should be multiplied by (1 + .0026 cos 2/), in which I is the lati-
tude, since the force of gravity varies according to this law.
Introducing these corrections into the formula and reducing, it
may be written in the following form,
E = 120(logp — log/) (502 + t + tf),
which may be applied directly to observations taken with an ane-
roid. For a mercurial barometer, p and p', must be corrected, first
for capillarity, and then divided by (1 + .00009 T) and (1 +
.00009 Tr). The correction for latitude is always small, and be-
comes 0 at 45°.
59. BUNSEN PUMP.
Apparatus. Fig. 46 represents a Bunsen filter-pump, such as is
used in chemical laboratories. A is" a valve in the supply-pipe, by
which the water is admitted to the bulb IB. From this it with-
draws a portion of the air, which passes down the pipe E with the
water. The vessel to be exhausted (7, is connected with B by
the long pipe (7Z?, through which the air is drawn. Above G is
placed a U mercury-gauge, and below it a wide tube, designed to
prevent the pressure from exceeding a certain amount. A fine
hole is made near the bottom of this tube, and it dips into a mer-
cury-cistern D. As the pressure diminishes, the mercury rises in
D and falls in the outer vessel until below this hole, and the air
rushes in and increases the pressure ; by varying the height of
the cistern any pressure may be maintained. This device, though
excellent in theory, often gives trouble in practice from the jump-
ing of the mercury, unless the tube is large and the hole small.
Instead, therefore, two or more valves may be used, or the tube
nearly closed, and thus the air admitted so slowly as to keep up
the required pressure. If too much water is passed through B it
sometimes overflows into O. An arm and stop should therefore
be attached to A, so that it cannot be opened too far. The
water escaping from E is received in a Florence flask, F, which is
fitted with a second tube G, passing nearly to the bottom, while
E opens near the top. To measure the amount of water expended,
a balance and weights should be provided, or a large graduated
vessel.
Experiment. Open A and the water will flow through .Z?, and
there encountering the air, will carry it in bubbles through E. If
BUNSEN PUMP.
119
Fig. 46.
now C is closed, the air will gradually be carried out of J9, pro-
ducing a rarefaction, and the air-bubbles
in E will be found to occupy less and less
space compared with the water, until the
limit of exhaustion is reached, and the
tube carries off nothing but water. The
diminution in pressure thus obtained
should nearly equal that of a column of
water of height BE, or if this is made
40 feet, nearly all the air should be with-
drawn. The aqueous vapor, however, al-
ways remains, and for other reasons the
exhaustion is never perfect; it neverthe-
less forms a very convenient method of
producing a partial vacuum.
To test the working of the pump and its efficiency, the follow-
ing experiment should be performed. Pour water through G until
F is filled up to the end of tho tube E. Empty it by blowing
through E, collect the water escaping from 6?, and weigh it. The
weight in grammes gives V^ the volume in cubic centimetres of the
portion of F included between the ends of E and G. Measure
the temperature of the water and the height of the barometer. Fill
F as before, take C out of the mercury, and open A slightly. A
large amount of air and a small amount of water will now enter the
flask. Water will flow from G until a volume of air equal to V
has entered the flask, and air begins to bubble up through G.
Collect the water that has escaped, and weigh it. Calling its vol-
ume V, the amount of water brought from B is evidently V —
V, while in the same time "F centimetres of air have been brought
down. Record also the time required to empty F. Repeat the
experiment several times with a larger flow of water. Try also
the effect of a flow under pressure by connecting the end of C in
the mercury, which will then 'rise in it when the water is turned
on, and may be kept at any desired height by raising or lowering
D. As in these experiments it takes some time for the mercury
to attain its normal level, it is well to connect a third tube with the.
flask, which may then be filled without disconnecting it from JS.
It will be seen that the best results are attained when the smallest
120 AIR-METER.
amount of water is used, but as the exhaustion then takes place
very slowly, it is often best to begin with a large flow, and dimin-
ish it as the air is withdrawn. The maximum amount of air that
might be drawn out by the apparatus may be determined analyt-
ically, and dividing the observed amount by this, gives the effi-
ciency.
This same apparatus may also be employed with advantage, to
test aneroid barometers. G is attached to an air-tight chamber,
formed of a tubulated receiver placed on an air-pump plate. When
the water is turned on, the air is gradually withdrawn, and the
barometer falls. The reading is compared with the true pressure
found by subtracting the reading of the U gauge from the height
of the standard barometer. Different results will be attained ac-
cording as the barometer is placed vertically or horizontally, or
if the friction is reduced by gently tapping on the instrument.
To render the test more complete this experiment should be tried
at different temperatures, which is best effected by a water jacket,
which may be filled either with hot or cold water.
60. AIR-METER.
Apparatus. An organ bellows, such as is described in the next
experiment, and an air-meter, of which a very convenient form is
that manufactured by Casella. It consists of a very light fan-
wheel, like a wind-mill, with a counter to record the number of
revolutions. The vanes are set at such an angle that the divisions
of the dial shall represent the number of feet traversed by the
air.
Experiment. Work the bellows and allow the air to escape
through an orifice, so as to produce a constant current of air.
Measure its velocity at intervals of ten inches from the orifice,
until it becomes imperceptible. Measure also the velocity on each
side of the central line. A spring catch serves to throw the gear-
ing in or out of connection with the fan-wheel. To make an
observation, therefore, the meter should be placed in the current,
and when it has attained a uniform velocity thrown in gear for
'exactly one minute. As the hands move only during this time,
the difference of readings taken before and after, give the distance
traversed, or dividing by 60, the velocity of the air per second.
AIR METER. 121
A table of corrections accompanies the instrument, showing how
much should be added or subtracted from the observed readings
to get the true distance.
Interesting results may be attained with this instrument on the
velocity of the wind, especially during gales, the air currents in
buildings from registers, ventilators or doors slightly open. This
forms one of the most efficient means of studying the ventilation
of large halls and churches.
SOUND.
61. SlRENE.
Apparatus. An organ bellows capable of giving a perfectly
constant current of air under various pressures. One of the best
forms is that made by Cavaille Coll (sold by Konig) with regu-
lator attached. If preferred, a large gas-regulator may be at-
tached to any bellows. A set of organ-pipes well tuned, giving
the notes of the scale from C3 to C"4, two or three tuning forks,
one giving the French normal pitch, etc. and a sirene. The lat-
ter need not be of large size, as good results may be obtained
with a single moving disk with one circle of holes.
Experiment. Place the organ-pipe, <73, in its hole on the bel-
lows, and connect the sirene so that the air shall pass through it.
Work the bellows, and the perforated disk will begin to revolve, at
first slowly, giving a rustling or humming sound, and then faster,
producing a note of low musical pitch. As the speed increases
the pitch rises, until it is about that of the pipe. Sound the lat-
ter, and increase or diminish the pressure of the air, so that they
shall be precisely together. A slight deviation produces beats,
that is, an alternate increase and diminution in the intensity of the
sound for every vibration gained or lost by the sirene. By a little
practice these beats may be made to take place very slowly, or not
at all. The wheelwork of the sirene may be thrown in or out of
gear with the revolving shaft, so that the hands may or may not
register the number of turns of the perforated disk. Throw it
out of gear, and read the position of the hands. Bring the two
sounds in unison, and keep them together for a minute, during
which time the shaft is thrown in gear, and the hands are moving.
The difference of the readings before and after, gives the number
of turns, and this multiplied by the number of holes in the per-
forated disk, give the number of complete vibrations. Dividing
122
KUNDT'S EXPERIMENT. 123
by 60 gives the number per second. Repeat with the other pipes,
and see if this ratio is that given by theory. Do the same with
the tuning forks. This is more difficult as they cannot be sounded
continuously. The best method of sounding a tuning fork is by
means of a violin bow. The latter should be held near the end
of the fork, nearly parallel to the two prongs, but touching only
one, and drawn with considerable pressure, and not too rapidly
To prevent slipping it should be well rubbed with resin.
62. KTJNDT'S EXPERIMENT.
Apparatus. Several glass tubes two or three inches in diame-
ter, and six feet long, one open at both ends, the others closed and
filled with different gases, and also containing a little lycopodium
powder. A number of rods of brass, steel, glass and wood, and a
clamp by which they may be held at the centre. Three of the
rods should be of the same material, but one of double diameter,
the second half the length, of the third. Cloths which may be wet
or covered with resin should be provided to set them in vibration,
also some lycopodium powder.
Experiment. Place a little lycopodium powder in the open
tube, hold it horizontally by the middle, and rub it lengthwise
with a wet cloth. A clear musical note of high pitch is at once
produced, and the powder arranges itself in about fifteen to
twenty groups at regular intervals along the tube. The reason is,
that the air in the interior of the tube vibrates with the same
rapidity as the glass, but as the velocity of sound in it is much
less, the wave-length is less in the same proportion. Hence divid-
ing the length of the tube by the distance apart of the lycopo-
dium groups gives the relative velocity of sound in glass and air,
or multiplying this number by 33& gives the velocity of sound in
glass in metres.
If the tube is filled with any other gas than air the interval will
be proportional to the velocity. Thus knowing the velocity in
glass, the velocity in the gas may be obtained. Make this meas-
urement with the other tubes, and see if the law holds that the
velocity is inversely proportional to the square root of the density.
This same method may be applied to the accurate determination
of the velocity of sound in solids. One of the rods is clamped at
the centre, and the end inserted in the open glass tube. The air
124
MELDE S EXPERIMENT.
in the latter is confined by a cork at one end, and a disk some-
what smaller than the tube is attached to the rod. The latter is
now set in vibration by a cloth' moistened with water, for glass, or
covered with resin, for wood or metal. The vibrations of the rod
are transmitted to the air, and the heaps of sand formed. In
general these will not be clearly defined, because the whole length
of the air space is not an exact multiple of half a wave-length.
The rod should therefore be moved in or out until the heaps are
distinctly marked. The velocity of sound in the rod is then ob-
tained by the following .calculation. If L is the length of rod,
I the distance between the heaps of lycopodium, and V =
333 (1 + .0037 £)2, the velocity of sound in air at any tempera-
ture t, then the velocity in the rod = — j- . The temperature t
may be taken as equal to that of the room, and measured with a
Centigrade thermometer.
63. MELDE'S EXPERIMENT.
Apparatus. A tuning-fork projecting horizontally from a verti-
cal wall, and tuned to give a low note, as G^. Four weights in
the ratio 1, £, £, fy, made of brass rods cut to these lengths respec-
tively, some fine silk thread, and a millimetre scale. A piece of
brass with a hole in it should be fastened to the end of one prong
of the fork, and a fine wire hook attached to the silk to support
the weights. A violin bow is also needed to excite the fork, or
bette, ran electro-magnetic attachment, by which the vibrations
may be maintained continuously.
Experiment. By this apparatus the various laws for the vibra-
tions of cords may be proved. 1st. The time of vibration is
proportional to the length. Place the fork so that its two prongs
shall lie in the same vertical plane, and suspend the largest weight
from it by the silk thread. Sound the fork, as described in Ex-
periment 61, and vary the length of the thread until its time of
vibration corresponds with that of the fork. When this is the
case it will form a loop or spindle, fixed at the ends and swelling
out at the centre through several inches. As this occurs only
when the cord is very nearly the right length it may be tuned
quite accurately by the eye alone. Make three or four observa-
tions in this way, measuring the length in each case with the mil-
ACOUSTIC CURVES. 125
limetre scale. Next turn the fork 90°, so that the prongs shall lie
in the same horizontal plane. The cord will now make as many
vibrations as the fork, while in the former case it made but half as
many. This is evident if the relative positions of the prong and
cord are compared. When the prong is in its highest position the
cord is straight or central. As the prong descends it moves to
the right, and as it ascends again becomes central. At the next
descent of the prong it moves to the left, becoming central a second
time, when the prong has reached the top. It thus makes only one
complete vibration, while the fork makes ,two. Accordingly when
the fork is turned around 90°,£he cord will be set vibrating ex-
actly twice as fast as before. On trying the experiment it will be
found that the new length of cord required will be just one half
that in the first case. That is, a double length requires a double
time.
2d. The time of vibration is inversely proportional to the
square root of the tension. Applying the four weights in succes-
sion, the corresponding lengths are proportional to 1, 2, 3 and 4, or
since for equal tensions the times are proportional to the lengths,
the law is proved.
3d. The time is proportional to the diameter of the cord. A
second cord of precisely twice the diameter of the first, may be
made by twisting four strands of the former. It will be found
that the length must be reduced one half to obtain the same
effect as before.
All these laws may also be proved by preparing a string of such
a length that it will vibrate as a whole, when the larger weight is
applied, then attaching the other weights it divides into 2, 3 or 4
loops, separated by fixed points or nodes, and corresponding to the
harmonics of the cord. A second string of double thickness
serves to prove the 3d law. A simple proof of the first law may
also be obtained by a second fork an octave higher than the other.
64. ACOUSTIC CURVES.
Apparatus. In Fig. 47, A is a large tuning-fork capable of giv-
ing out at least one harmonic besides its fundamental note, and
carrying on the end of one of its prongs a piece of sheet brass cut
to a point. B is a little carriage on which a piece of smoked
JFD
126 ACOUSTIC CURVES.
glass may be laid and drawn under the brass point or style by a
cord passing over the pulley C. Two weights, D and E, are at-
tached below, the upper one, D, being just equal to the friction of
the carriage. Some pieces of glass about three inches by four, are
needed, and a gas-burner, by which they may be covered with
lampblack. By using the size of glass employed in the lantern
for projections, the curves may be thrown on the screen on a
greatly enlarged scale.
Experiment. Cover one of the plates of glass with a layer of
lampblack by holding it by one corner over the gas-flame, and
moving it about so that the
coating shall be uniform, and
very thin. Instead of lamp-
black, collodion may be used,
pouring it on in the usual
way, as when taking a photo-
graph. Care must be taken
Fis-47- to select such collodion as
will give an opaque and very tender film, when results of extreme
beauty and delicacy will be obtained. Lay the glass down on the
carriage, and raise it so that when passed under the style, "the
latter will just touch its surface. This may be accomplished by
wedges or levelling screws under the glass. Draw JB back a short
distance beyond the style, and release it, when it will begin to
move under the action of the two weights D and E. The length
of the cord should be such that when the wagon reaches the style,
E will touch the floor so that the carriage will move with a uni-
form motion by its inertia, the friction being just compensated by
D. The style will accordingly draw a fine unbroken straight line
over the glass. Now sound the fork by the violin bow (see Ex-
periment 61), and again pass the carriage under, when the line,
instead of being straight, will be marked by sinuosities, one cor-
responding to each vibration of the fork.
Next sound the harmonic, by drawing the bow somewhat more
rapidly, and with less pressure than before, at a point about two-
thirds of the distance from the end of the prong to the handle.
The sound sometimes comes out more readily by lightly touching
the intermediate one-third point or node with the finger. A high,
clear note is thus produced, and on drawing the carriage back the
ACOUSTIC CURVES. 127
same distance as before, and letting it again pass under, another
curve is obtained, with indentations much nearer together, owing
to the greater rapidity of the undulations. Of course the plate is
moved sideways a short distance each time, to prevent the curves
from overlapping. Produce the fundamental note, and while it is
sounding draw the bow so as to give the harmonic, and immedi-
ately let go the carriage. A curve is thus obtained, resulting from
these two systems of vibrations, and consisting of small sinuosities
superimposed on larger ones. Determine their ratio by seeing
how many of the former correspond to an exact number of the
latter. Write on the lampblack your name and the date, and if
all the curves are good, varnish the plates to render them perma-
nent. For this purpose expose the blackened surface to the vapor
of boiling alcohol to remove the grease, then holding it by one
corner pour amber varnish over it precisely as when varnishing a
photographic negative.
To compare the lines with theory, place the glass in a magic
lantern, and project an image of it on the screen. If the sun is
used as a source of light, it is scarcely necessary to darken the
room. Place a sheet of paper so that three or four undulations of
the curve of the fundamental note shall fall on it. Trace. them
carefully with a pencil and an enlarged reproduction of the origi-
nal is obtained. Draw lines tangent to the waves above and
below, and bisect the space between them by a line. It will
intersect the curve in points at regular intervals #, any one of
which may be taken as the origin of coordinates. If a is the
height of the wave, or one half the distance between the two
-~/y»
tangent lines, the theoretical equation will be y = a sin -r- . Con-
struct points of this curve by dividing the space between two
consecutive intersections of the curve into six equal parts, and lay
off vertical distances equal to a multiplied by sin 15°, 30°, 45°,
etc., to 180°. These sines have the following values: — sin 15° =
.259, sin 30° = .500, sin 45° = .707, sin 60° = .866, sin 75° = .966.
Draw a smooth curve through the points thus obtained, and com-
pare it with that given by the forks. To test the combination of
the two systems of vibrations is more difficult, but it may be done
128 LISSAJOUS' EXPERIMENT.
by taking their equations separately, yf — a sin -T-, and y" =
ar sin -^7, and adding them so that y =• y' + y" = a sin V +
a' sin ~.
An immense variety of curves may be obtained by mounting
the plate on a second tuning-fork, which is also set vibrating.
Different curves are thus obtained, according as the motion of the
style is parallel or perpendicular to the vibrations of the plate,
also with every change in the interval between the two forks.
With this arrangement it is much better to maintain the vibra-
tions of one or both forks continuously by electricity. Better
effects are also obtained in this way in Melde's and Lissajous' ex-
periments.
Instead of projecting the curve on the screen it may be measured
by the Dividing Engine, Experiment 21, or enlarged by a micro-
scope and drawn by a camera lucida, The length of the waves
gives a very delicate test of the uniformity of the motion of the
car, a difference of a ten thousandth of a second being easily
perceived.
65. LISSAJOUS' EXPERIMENT.
Apparatus. Mirrors are attached to the ends of the prongs of
two tuning-forks, and the image of a spot of light reflected in
them is viewed in a telescope. The planes of the tuning-forks
must be perpendicular, that is, one must vibrate in a vertical, the
other in a horizontal plane. It is best to have a series of forks
with sliding weights, so that all the intervals in the octave
may be obtained. A good spot of light is produced by a gas flame
shining through a small aperture in a metallic plate, or a mirror
may be used to reflect the light of the sky.
Experiment. On looking through the telescope a minute spot
of light should be visible. When one of the tuning-forks is
sounded the mirror is moved from side to side, carrying the image
of the spot with it so rapidly as to make it appear like a horizon-
tal line of light. In the same way the motion of the other fork
produces a vertical line. When both sound, a curve is formed,
which remains unchanged if the concord is exact, but continually
alters if the forks are not a perfect tune. Bring the forks in uni-
LISSAJOUS' EXPERIMENT. 129
son by placing the weights on the corresponding points of each.
They are best sounded by a bass-viol bow, drawing it slowly and
with pressure over the end of one prong nearly parallel to, but not
touching, the other. As the bows soon wear out by the horsehair
giving way, a convenient and cheap substitute is made by covering
a strip of wood of proper shape with leather, which when rubbed
with resin, answers very well.
On sounding both forks, having brought them in unison as
above, the point of light is in general converted into an ellipse
which, as it is impossible to tune them exactly by the ear, grad-
ually changes into a straight line, then into an ellipse, a circle, an
ellipse turned the other way, a straight line and so on. Raise the
pitch of one of the forks slightly, by moving the weight towards
the handle, and if the changes take place more slowly the unison
is more perfect. By trial, first moving the weights one way and
then the other, they may be brought in tune with any desired
degree of exactness, and far nearer than is possible by the ear
alone, as the complete change of the curve from one line to the
other denotes that one fork has advanced only a single vibration.
Next make one fork the octave of the other, and a curve is ob-
tained, changing from the parabola to the lemnescata, or figure 8.
A simple rule serves to determine the interval in all cases from the
curve. Count the number of points where the latter touches the
sides of the rectangle bounding it, also the number of points
where it touches its top or bottom ; the ratio of these two is the
interval between the forks. When the curve
terminates in either corner this point must be
counted as one half on the horizontal, and half
on the vertical, bounding line. Thus in Figure
48, both A and B correspond to the ratio of
2 : 3, or the interval of the fifth.
The more perfect the concord the more slowly will the curves
alter their form, and the simpler the ratio of the number of vibra-
tions the simpler the curve. When the forks are not quite in
unison, beats will be heard, and the curve will then be seen to
alter its form so as to keep time with them. Next try some other
ratios, as f , f, f , & 4» i > also some more complex curves, as f , f ,
and -.
130 CHLADNI'S EXPERIMENT.
66. CHLADNI'S EXPERIMENT.
Apparatus. A number of brass plates attached to a stand, a
violin bow and some sand. A good series of plates consists of
three circles, whose diameters are as 2 : 2 : 1, and their thickness
as 1 : 2 : 1. Also three square plates, similarly proportioned.
^Experiment. The plates are sounded by touching them at cer-
tain points and drawing the bow across their edges, holding it
nearly vertical, and moving it slowly and with considerable pres-
sure. A sound is thus produced, and certain lines are formed on
the plate called nodal lines, which remain at rest, the other parts
vibrating. If sand is sprinkled uniformly over the plate, that on
the nodal lines will remain there, the rest being thrown up and
down, so that finally it will all collect on these lines. The higher
the note the more complex the nodal lines, and the nearer they
are together.
Taking first the largest and thinnest circular plate, touch it at
any point of the circumference, and bow it at a point about 45°
distant. The sand will collect on two lines at light angles. Next
bow it at a point 90° distant, and it will divide into six parts. By
touching the plate at two points distant 45° with the thumb and
middle finger of the left hand, and bowing the point midway be-
tween them, a division into eight equal parts is obtained. In the
same way 10, 12, or any even number of parts are formed, until
the divisions become so small that they cannot be sounded.
Next try the first square plate. The lowest sound this will
give is obtained by touching the centre of one side, and bowing
the corner. The next note, a fifth above, is produced, when the
corner is held and the centre bowed. By altering the position of
the fingers and bow, a great variety of figures may be obtained,
which may be still further extended by changing the points of
support, or the form of the plate. Moreover, among plates of the
same shape some seem to give out certain curves more easily
than others, owing probably to peculiarities in their internal struc-
ture. A square plate generally gives readily, besides the curves
described above, one formed of two diagonal lines, and four half
ovals on its edges. The pitch is three octaves above that of the
diagonal lines alone. Another curve of extreme beauty consists
of a circle with eight radial lines, and the intermediate spaces
CHLADNIS EXPERIMENT. 131
marked by eight half ovals. It will be noticed that the points
touched are necessarily at rest, and hence lie on the nodal lines.
By using the three plates of the same form, some of the laws of
the vibrations of plates may be proved. 1st. The number of
vibrations is proportional to the thickness. Form the same curve
on the two plates, of which one is double the thickness of the
other, and it will be noticed that the pitch is always an octave
higher. 2d. In similar plates of equal thickness, the number of
vibrations is inversely as the square of the homologous parts.
Hence the small plate gives a note two octaves above the large
one of the same thickness.
LIGHT.
67. PHOTOMETER FOR ABSORPTION.
Apparatus. In Fig. 59, A is the source of light, which may be
a candle in a spring candle-stick, or a small gas jet. B and C are
two mirrors set at such an angle that they will form images of A,
just 50 inches apart, and making ABC a right-angled triangle. D
is a Bunsen photometer disk, made by placing a circular piece of
thick paper in a lathe, and painting all but the centre with the
best melted sperm-candle wax. It then possesses the property
when placed between two lights, of changing its appearance ac-
cording as one or the other is the brighter. It is mounted on a
slide and carries an index which moves over a graduated scale.
F is a screen so placed as to protect D and the eyes of the ob-
server from the direct light of A, while it leaves the scale illumin-
ated so that it can be easily read. A stand with a graduated
circle is also provided, on which one or more plates of glass may
be set and inclined at any angle to the ray of light AJB. Some
observers prefer a disk with only the central part covered with
wax, and instead of a circular spot use some other form. The
great difficulty in these cases is to distribute the wax uniformly,
and prevent its accumulating at the edges. Still another method
is to punch figures in a sheet of thick paper, and cover both sides
of it with tissue paper, taking care that no wrinkles remain. The
whole apparatus must be used in a darkened room.
Experiment. The disk D possesses the curious property of
appearing bright in the centre when
a strong light is in front of it, but
dark when the brighter light is be-
hind. When placed between two
lights there is therefore a certain
position where the spot will disppear,
in which case it is so much nearer
the fainter light that the illumination
on its two sides are equal. Their rel-
ative brightness may then be readily
132
PHOTOMETER FOR ABSORPTION. 133
determined from the law that the intensity is inversely as the square
of the distance. Now the two images of A act like two precisely
similar lights, any change in one affecting the other equally. By
moving D the centre spot may be made either light or dark, and
there will be a certain intermediate position in which it will dis-
appear. The exact point of disappearance can be determined only
by long practice, noticing that it varies with the position of the
eye, and with the two sides of the disk. Find this point as nearly
as possible, read the index, move D a short distance, set again
and take the mean of several such observations. Compute
the probable error in inches, and the result multiplied by 4
gives the error in percentage. Let x be the mean observed
reading, or AB + BD. Then the distance of the other image of
the light equals DC + OA^ or 50 — x. Calling B and O their
intensities at a distance unity, their intensities at the distance of
7? C1
the disk will be —3 and ,*„ — c-2 ; or since these quantities are
B / x \2
equal, their relative intensities I = -^ = ( ^-— — i . Next place
a piece of plate glass carefully cleaned on the stand between A
and B, and at right angles to the line connecting them. To make
this adjustment remove D, and place the eye beyond the light.
Then turn the stand until A^ its reflection in the plate glass, and
its reflection in B and (7, all lie in the same straight line. Now
set the disk as before, record the mean reading, and compute the
relative intensities. Increase the number of plates one at a time,
and compute the intensity in each case. This number divided by
that when no plates were interposed, gives the percentage trans-
mitted.
This same apparatus is well adapted to determine the amount
of light transmitted at different angles of incidence, that cut off
by ground glass, the effect of the snuff of a candle, or the over-
hanging portion of the wick, and the comparative brilliancy of
the edge and side of a flat flame. In the last case it is only nec-
essary to set the flame so that it shall shine edgewise, first into B
and then into (7, and compare the position of the disk in the two
cases. It will thus be found that the prevalent impression that
flame is perfectly transparent, is erroneous.
184 DAYLIGHT PHOTOMETER.
68. DAYLIGHT PHOTOMETER.
Apparatus. In Fig. 50, AB is a box about six feet long, a foot
wide, and a foot and a half high. It may be made of a light
wooden frame covered with black paper or cloth. A circular hole
about four inches in diameter is cut in the end B, and covered
with blue glazed paper with the white side out, and made into a
Bunsen disk by a drop of melted candle-wax in the centre. A
long wooden rod rests on the bottom of the box, and has a stand-
ard wax candle, A^ in a spring candle-stick attached to one end.
The distance of the candle from the disk may thus be varied at
will, and measured by a scale attached to the rod. The box
should be ventilated by suitable holes cut in it, or the air will
become so impure that the candle will not burn properly.
Experiment. This instrument is intended to compare the
amount of light in different portions of a room, or its brightness
at different times.
When the candle is
placed at a distance
0 . from the photome-
ter disk, the latter
will appear dark in
Fi 50 the centre, while by
making AB very
small, so that the strongest light shall be inside, the centre will be
bright. The color of the candl'e flame being of a reddish tint
compared with daylight, is first passed through the blue paper,
which thus renders the colors more nearly alike. When the dis-
tance of the candle is such that the illumination is equal on both
sides of the disk, the spot will nearly disappear, and unity divided
by the square of this distance gives a measure of the comparative
brightness under various circumstances.
An excellent experiment with this instrument is to measure the
fading of the light at twilight. Light the candle and place it at
such a distance from the disk that the spot shall disappear, as in
the last experiment. As the light diminishes, the distance AB
must be increased. Take readings at intervals of one minute, and
construct a curve with ordinates equal to one divided by the
square of this distance, and abscissas equal to the time. The
amount of light for different distances of the sun below the hori-
BUNSEN PHOTOMETER. 135
zon may be obtained directly from this curve. In the same way
the brightness of different parts of the laboratory may be meas-
ured, the effect of drawing the window curtains, and the compara-
tive brightness of clear and cloudy days. This apparatus was
used during the Total Eclipse of 1870, to measure the amount of
light during totality, possessing the advantage that on returning,
the precise degree of darkness could be reproduced artificially.
Comparisons may also be made with moonlight, the light of the
aurora or other similar sources of light.
69. BUNSEN PHOTOMETER.
Apparatus. A photometer room forms a most valuable addition
to a Physical Laboratory, both on account of the great variety of
original investigations which may easily be conducted in it by
students, and also owing to the practical value of the instrument,
and the excellent training it affords in the use of various forms of gas
apparatus. If a separate room cannot be obtained, a part of the
laboratory may be partitioned off by paper or cloth screens black-
ened on the interior, so as to leave a space about twelve feet long
by five wide and eight feet high, which should be nearly dark, and
supplied with some means of ventilation. In this is a table ten
feet long, a foot and a half wide, and three high, and over its cen-
tre, at a height of five feet from the floor, the photometer bar, AB,
Fig. 51, is placed. The latter is 100 inches in length, and divided
on one side into inches and tenths, and on the other into candle
powers. To make this graduation, calling x the distance from one
end of the bar in inches, and C the corresponding candle power,
we have, as will be seen below, x2 : (100 — xY = C : 1, or x =
*J C
100 1 -L. / Q- By making C = 1, 2, 3, etc., the bar may be grad-
uated as desired. At one end of the bar is placed a sperm candle,
-4, supported in a balance for determining its loss of weight as it
burns. The best form is that invented by Prof. F. E. Stimpson,
on the principle of the bent-lever balance, in which the motion of
a long arm over a scale shows the number of grains consumed.
At the other end of the bar, gas is admitted, and its brightness
when burned compared with that of the candle. The pipe sup-
plying the gas passes through the meter F, which is of the form
known as the wet meter, and indicates the volume to one thou-
sandth of a foot. To read it, a separate burner E is provided,
supplied with gas, which does not pass through the meter. Of
course it is turned down when setting the disk. The gas passes
from the meter to the regulator G, by which the pressure is ren-
dered perfectly constant. This consists of a bell resting in water,
like a gas-holder, with a long conical rod attached to its centre,
136
BUNSEN PHOTOMETER.
which, when raised, cuts off the supply of gas. Whenever the
pressure becomes too great, the bell rises and reduces the flow of
gas, while too small a pressure makes the bell descend and admit
more gas. Beyond the regulator a f is inserted, and connected
with a gauge I, which gives the pressure. In its simplest form
this is a bent tube containing water, which should be of consider-
able size, if accuracy is required. Sometimes a floating bell is
used, which rises and falls as the pressure varies, and moves a long
index over a graduated scale. The gas next passes to the burner
B, first traversing one or two stopcocks, H, to regulate the quan-
tity consumed. A variety of burners should be procured, which
may be used in turn and compared.
A slide G is placed on the photometer bar, carrying a Bunsen
photometer disk (Experiment 67). When this is placed between
two lights, if the brightest is in front, the small circle looks light
on a dark ground, if the brightest is behind, it appears dark.
A clock D, marking seconds, is also needed in this room, the
best form striking a bell at the beginning of each minute, also five
seconds before it, and having what is called a centre seconds' hand.
It is often convenient to have a separate gas-pipe, meter and
burner at A, the candle-end of the photometer bar, and to have
the latter arranged so that it can be swung horizontally to it. In
fact, in almost every new research some change of arrangement
will be found desirable, and the apparatus should therefore not be
fixed, but arranged so that the connections can be easily altered.
Experiment. To measure the candle power of burning gas.
The law in the State of Massachusetts requires that the gas fur-
Fig. 51.
nished, when burnt at the rate of five feet per hour, under the
most favorable circumstances, shall give a light at least equal to
BUNSEN PHOTOMETER. 137
twelve sperm candles (6 to the pound), when consuming 120
grains per hour. To make the experiment the gas and candle are
burnt at opposite ends of the photometer bar, their relative inten-
sities compared, and the consumption of each measured. The
amount of wax burnt is measured by the candle-balance A. This
consists of a sort of steelyard, with a light weight or rider K,
moving over its longer arm, which is divided so as to give grains.
The centre of gravity of the beam is at such a distance from the
point of suspension that the sensibility shall not be very great, and
an index is attached, which moves over a scale, each of whose
divisions corresponds to a change in weight of one grain.
Attach the candle to the shorter end of the beam and light it ;
set the rider at zero, and place weights in the scale-pan, until that
end of the balance is somewhat the heaviest. Now as the candle
burns it becomes lighter, and soon begins to rise, its diminution
grain by grain being shown by the index moving over the scale.
Light the gas also at the other end of the beam, and place the
photometer-disk on the bar ready for use. Precisely at the begin-
ning of a minute, as given by the clock, read the gas-meter, re-
cording the feet and thousandths, also the position of the index of
the candle-balance. The observations commonly extend over five
minutes, and in this time 10 grains of wax should be consumed •
set the rider therefore at 10, and if the candle is burning at the
standard rate, the position of the index at the end of that time
will be the same as at the beginning, if not, the difference shows
the correction to be applied.
Next measure the intensity of the two lights by the photometer
disk. As stated above, this possesses the property of appearing
light on a dark background, or the contrary, according as the
brightest light is in front of, or behind it. By moving it back-
wards and forwards therefore, a point will be found where the
spot will disappear almost completely, owing to the equality of
the two lights. Read its position by the graduation and record,
then move it, and set again several times. At the end, of the min-
ute read the meter, and then take some more readings of the disk.
Try also setting the disk so that the spot shall be first slightly
brighter, and then equally darker than the adjacent paper. This
is called taking limits, and the mean gives the true reading. Pro-
138 LAW OF REFLECTION.
ceed in this way for five minutes, reading the meter at the end of
each minute, and taking two or three intermediate settings ol the
disk. At the end of the time read the index of the candle-balance
also.
From these data the candle power may be computed as follows.
The consumption of the candle is obtained by subtracting the first
reading of the index from the last, and adding the difference to
10. This gives the consumption in 5 minutes, and multiplying it
by twelve gives (7, the number of grains per hour. It is, however,
safer to extend the reading of the candle-balance to a longer time,
as fifteen or twenty minutes, to diminish the errors. The con-
sumption of gas per minute is obtained by subtracting each read-
ing of the meter from that which follows it, and multiplying by
6Q gives 6r, the rate per hour. Call I the ratio of the two lights,
as given by the mean of the readings of the scale attached to the
bar, and apply the following corrections. First, for rate of candle^
it is assumed that the light is proportional to the consumption.
r<
Hence the corrected candle power _Z7 : L = C\ 120, L' = L ~^-
Again it is assumed that the light of the gas is proportional to its
C*
consumption, or to -JT-, and dividing by this fraction gives what
would be the candle power if just 5 feet were burned, or the true
5 (75
candle power L" = L'~Q \ hence L" = L • J^Q • -g
This example serves to show how the photometer is ordinarily
used, but it may be applied to a great variety of investigations.
For instance, different burners may be compared, or a single
burner under varying consumption. The amount of light cut off
by plain and ground glass at various angles may be measured,
and the effect of changes in moisture, in temperature, or in
.barometric pressure studied.
70. LAW OF REFLECTION.
Apparatus. In Fig. 52 a circle divided into degrees is attached
to a stand, and carries two arms with verniers, or simple pointers,
C and D. The first is attached to a centre plate, which carries
a vertical mirror placed at right angles to BC. This mirror
is silvered on its front surface, or may be made of blackened glass,
B
A
ANGLES OF CRYSTALS. 139
and a vertical line is ruled on it, which is brought to coincide ex-
actly with the centre of the circle. A vertical rod or needle is
attached to J9, whose reflection in B is to be observed at different
angles of incidence. A is a piece of brass with a small hole
pierced in it to look through.
Experiment. Bring O in line with B and A> and turn it so
that on looking through the latter the reflection of the hole may
be brought to the centre of the mir
ror and bisected by the line marked
on it. The reading of the index C
gives the zero, or starting point.
This observation should be repeated
two or three times, dividing the de-
grees into tenths by the eye. Turn
C a few degrees and bring D into such a position that the reflec-
tion of its needle shall coincide with the line on C. Now the
difference in reading of D and (7 will equal the angle of incidence,
and the difference between the reading of C and the zero equals
the angle of reflection. By the law of reflection these two angles
should be equal. Repeat this observation with different parts of
the graduated circle, at intervals of about fifteen or twenty de-
grees. Small deviations from the law serve well to exemplify the
different kinds of errors of observations. Thus if the needle is
not exactly on the line connecting its index with (7, a constant
error will be introduced. If the mirror is not exactly over the
centre of the circle, the difference will vary in different parts of
the circle, causing a periodic error. If the differences between the
angles of incidence and reflection are sometimes positive and
sometimes negative, they are probably due to accidental errors,
such as errors in graduation, in reading, unequal fitting of the
parts, etc. Finally, if a single observation gives a large error, it is
probably due to a mistake, or totally erroneous reading.
71. ANGLES OF CRYSTALS.
Apparatus. The instrument most commonly employed to
measure the angles of crystals is Wollaston's reflecting goniometer,
represented in Fig. 53. A is a vertical circle divided into degrees,
and turned by a milled head B through any given angte, which is
measured by a vernier C. A second milled head D is attached to
140 ANGLES OF CRYSTALS.
a rod passing through the axis of this circle with friction. The
crystal is fastened by wax to a small brass plate E, bent at right
angles and resting in the split end of the pin F. By this it may
be turned horizontally, and the joint Gr gives a vertical motion.
The whole is mounted on a stand, which should be placed on a
table opposite the window, and ten or twelve feet from it. A
spring stop is attached to the stand, and a pin placed in the grad-
uated circle, so that when the latter is turned forward it cannot
pass the 0°, or 180° mark, but may be brought by the milled head
exactly to this point. Several crystals to be measured should be
provided, some, as quartz, galena, alum or salt, well formed and
polished, and therefore easily measured, and others of greater diffi-
culty. The best material with which to attach them to E is a
little beeswax.
Experiment. The crystal must be fastened to the stand in such
a way that the edge to be measured shall lie exactly in the axis of
the instrument prolonged, and the main diffi-
culty in the experiment is to make this ad-
justment with accuracy. Attach the crystal
to the plate E by a little piece of wax, and
adjust the edge as nearly as possible by the
eye, turning it horizontally by the pin F, and
vertically around the joint G. It is thus
brought parallel to the axis, and may be made
to coincide with it by sliding the plate in the
Fig. 53. pm fi Select now two parallel lines, one of
which may be a bar of the window, and the other the further edge
of the table, or a line ruled on paper, and set the axis of the instru-
ment parallel to them. On bringing the eye near the crystal an
image of the window will be seen reflected in one of its faces, and
by turning either milled head the image of the bar may be brought
to coincide with the second line. If they are not parallel it shows
that the face is not parallel to the axis of the instrument, and the
crystal must be moved. Do the same with the other face to be
measured, and when both images are parallel to the line on the
table, both faces, and consequently their intersection, are parallel* to
the axis. This adjustment is most readily made by placing one
face as nearly as possible perpendicular to the pin F, when the
image in this face may be rendered parallel by turning 6r, that in
the other by turning F.
ANGLE OF PRISMS. 141
Now turn B until the circle stops at 180°, and turn D until the
image in the further face of the crystal coincides exactly with the
line on the table. Then turn B in the other direction until the
second image coincides, when the reading of the vernier will give
the correct angle. Evidently the two faces are in turn brought
into exactly the same position, and the angle between them equals
180° minus the amount through which the circle has been turned.
It is sometimes more accurate, though a little more troublesome, to
turn the crystal into any position by J9, and bring first one image
to coincide, and then the other. 180° minus the difference in the
readings of the vernier give the required angle. Try this with
different parts of the circle. Remove the crystal, attach it a sec-
ond time to E, and see if the same result is attained as before.
Repeat until readings are obtained differing from each other but a
few minutes. Also measure some of the crystals less highly
polished. An excellent test of the work is to measure the
angles completely around a crystal, and see if their sum equals
180° (n — 2), in which n is their number.
72. ANGLE OF PKISMS.
Apparatus. One of the most valuable instruments in a Physi-
cal Laboratory is the Optical Circle, or Babinet's goniometer.
This instrument may be used as a goniometer for measuring the
angles of crystals, to find the index of refraction of liquids or
solids, to study dispersion, or, as a spectrometer, to measure wave-
le/ngths. It is therefore often desirable to duplicate it, or perhaps
better, to procure one large and very accurate instrument, and
others of smaller size for work requiring less precision.
This instrument, Fig. 54, consists of a graduated circle on a
stand, with two telescopes, A and B, attached to it. A is the col-
limator, or a telescope in which the
eye-piece is replaced by a fine slit,
whose width may be varied by a screw-
resting against a spring, and whose
distance from the object-glass may be
altered by a sliding tube with a rack
and pinion. This telescope is at-
tached permanently to the stand, while
B, which is a common telescope with
cross hairs in its focus, is fastened to
an arm revolving around the centre of Fi£- 54-
the graduated circle. It may be held in any desired position
by a clamp D, moved slowly by a tangent screw E, and the angle
142 ANGLE OF PRISMS.
through which it has been turned, accurately measured by a
vernier. To eliminate errors of eccentricity a second vernier is
sometimes placed opposite the first, in which case the mean of
their readings is always employed. For great accuracy a spider-
line micrometer should be attached to J5 to measure small angles,
as will be described more in detail in Experiment 77. It is often
convenient to have both telescopes mounted on conical bearings
so that they may be turned away from the centre of the circle
when desired. They should also be supported in such a way that
one end of each may be raised or lowered a little, so as to bring
their axes perpendicular to that of the instrument. This is most
readily accomplished by placing an adjusting screw under one of
the Y's carrying them. C is a small circular stand on which prisms
may be placed, and which may be turned around the centre of the
circle and clamped in any position. Sometimes an arm and ver-
nier is attached to measure its angular motion, but this is not ab-
solutely necessary. Its principal use is to measure the angle of
crystals, and by it the law of reflection may also be proved with
great accuracy. The graduated circle is sometimes made to re-
volve, and the angle measured by one or more fixed verniers.
The whole is commonly mounted on a tripod with levelling
screws, as shown in the figure. These are ornamental rather
than useful, however, as in common experiments it makes no dif-
ference, except in appearance, if the circle is not properly lev-
elled. In any case, except to raise or lower the instrument, only
two screws are needed, and the third may be
— replaced by a fixed point. In this, as in all
^ instruments mounted on three legs, the best
form of support is that represented in Fig. 55.
One leg rests in a conical hole A, a second
in a wedge-shaped groove B, and the third on
9 a plane surface C. A fixes the position of
the tripod, which is prevented from turn-
Fig 55 *n& ky the groove .7?, while if the three legs
change their relative positions, IB can slide
back and forth in its groove, and C move freely over the plane
surface. If instead, three conical holes were used, and these were
not precisely in the right position, or the distance of the legs
varied with changes of temperature, the whole instrument might
be so strained as to introduce serious errors in the graduated
circle. This instrument should be placed near the window so that
sunlight may be reflected through it by means of a mirror, or if
preferred, the light from an Argand or Bunsen burner employed.
One or more flint glass prisms are also needed, all three of whose
faces should be polished and inclined at angles of 60°.
Experiment. The following adjustment must always be made
when the optical circle is used. Draw out the eye-piece of .Z?,
ANGLE OF PRISMS. 143
Fig. 54, until the cross-hairs are seen with perfect distinctness.
Then turn the telescope towards some distant' object and focus it,
moving both eye-piece and cross-hairs. Now both the object and
cross-hairs should be perfectly distinct, and not change their rela-
tive positions as the eye is moved from side to side so as to look
through different portions of the eye-lens. Sometimes the objective
alone moves, and sometimes the distance is permanently fixed, so
that it is in adjustment for parallel rays. Now turn the two teles-
copes towards each other and illuminate the slit either by placing
an Argand burner behind it, or reflecting the light of the sky
through it by means of a mirror. On looking through the ob-
serving telescope an image of the slit will now be visible. Focus
it, moving it towards or from its objective, when its distance will
equal the principal focal distance of the collimator, and the beam
of light between the two telescopes will be parallel, or as if coming
from a slit placed at a very great distance. Bring the image of
the slit to coincide exactly with the vertical cross-hairs in B by
the tangent screw, first champing the telescope. If it is not verti-
cal the slit may be turned, and if it is too high or too low it should
be brought to the centre of the field by raising or lowering one
end of one of the telescopes, as described above. Having ren-
dered the coincidence exact, read the vernier and repeat the
setting two or three times, as it gives the zero from which most of
the following measurements are made.
To measure the angle of a prism, stand it on the centre-plate
with its edges vertical, and with the faces whose angle is to be
determined about equally inclined to the axis of the collimator.
To eliminate parallax in case the telescopes have not been accu-
rately focussed for parallel rays, it is better to place the edge of
the prism over the centre of the graduated
circle. To 'prevent motion of the prism
when IB is turned, C should be clamped.
Let AJ3C, Fig. 56, represent the prism
whose angle A is to be measured, and DDf
the axis of the collimator prolonged. Turn
the observing telescope into the position
AF, when an image of the slit will be seen
on looking through. Bring it to coincide
144 ANGLE OF PRISMS.
with the cross-hairs by the tangent-screw, first clamping the teles-
cope, and read the vernier. Then turn the telescope into the
position AE and set again. The difference in the readings di-
vided by two, equals the angle of the prism. For D 'A C equals
90° — the angle of incidence, and FAC, 90° — the angle of re-
flection ; hence they are equal. In the same way, EAB = D' AB,
or FA C + EAB = BA C, and FAE — IB A C. Move the prism
a little, repeat the measurement, and see if the same result is
obtained as before. Determine in the same way the three angles
of the prism, and their sum should equal 180°. If either of the
reflected images of the slit is too high or too low, the base of
the prism is not perpendicular to the edges. In this case it must
be adjusted by placing pieces of paper, or tinfoil, under one or two
of its corners, until both images are in the centre. If either is
out of focus when the telescopes have been adjusted for parallel
rays the reflecting surface is curved instead of plane, while a dis-
tortion of the image shows that the surface is irregular. In either
case, an accurate measurement is impossible, since the angle will
vary for different parts of each face.
By the plan just described, the angle of a prism may be found
if, as is often the case, the centre plate has no vernier attached to it.
With such a vernier, however, the angle may be determined more
readily, as follows. Set the telescopes nearly at right angles, and
stand the prism on the centre-plate, as before, with its faces verti-
cal, and the edge to be measured over the axis of the instrument.
Turn the centre-plate until one of the faces is equally inclined to
the axes of both telescopes, when the image of the slit reflected in
this face will be seen in the field on looking through the observing
telescope. Bring it to coincide with the cross-hairs by the clamp
and tangent-screw, and read the vernier. Turn the centre-plate,
taking great care not to disturb the position of the prism on it,
until the image reflected in the other face coincides with the cross-
hairs. 180° minus the difference in the readings of the vernier
gives the angle of the prism. Repeat as before, and also measure
the three angles and see if their sum equals 180°.
When a vernier is attached to the centre-plate this instrument
serves to prove the law of reflection with great exactness. For
this purpose it is only necessary to turn the centre-plate into vari-
LAW OF REFRACTION. I.
145
ous positions, bring the reflection of the slit to coincide- with the
cross-hairs of the observing telescope, and read the verniers at-
tached to each ; or in fact, to repeat Experiment 70, replacing the
sight-hole by the slit and collimator, and the needle by the ob-
serving telescope.
73. LAW OF REFRACTION. I.
Apparatus. In Fig. 57, DBCG is a tank, like that of an aqua-
rium, with the side BD of glass. Two horizontal scales are at-
tached to CG, one over the other, and the tank filled so that one
shall be seen above, the other below, the liquid. A is a plate of
brass with a vertical slit in it, larger above, and tapering to a
point. It is used as a sight, and is placed at a distance AB equal
to BC. A small plumb-line may be hung in front of DB to serve
as an index. A tank without glass sides may be employed in-
stead, by regarding Fig. 57 as a vertical instead of a horizontal
section, and'placing one scale at the surface of the water, BD, the
other at the bottom, CGr. The divisions of the upper scale should
then be one half those of the lower.
Experiment. On looking through A*ihe lower scale will be
seen through the water, the upper through the air only. The
divisions of the former will therefore,
by refraction, appear larger than those
of the other, and from the amount of
this increase the law of refraction may
be deduced. Placing the plumb-line at
D, and looking through A, it will be
seen projected on the upper scale at
6r, but on the lower, owing to the bend- ^,:
ing of the ray at the surface BD, at !?•
If placed at B, however, the reading on
both scales will be the same, since the incidence being normal
there is no bending of the ray. To find this point, read both
scales, and if the reading of the upper scale is the greatest, move
B to the right, otherwise to the left, until both read alike. The
object of the varying width of the slit is to read approximately
through the upper part, and then lowering the eye to eliminate
parallax, and read more exactly by the lower portion. Move the
plumb-line a short distance, read both scales again, and thus take
ten or fifteen readings between B and D.
10
Fig. 57.
146 LAW OF REFRACTION. II.
ISTow from these readings to prove that "the ratio of the sines of
the angles of incidence and refraction is always constant and
equal to the index of refraction. In the figure, the angle of inci-
dence equals 90° — ADB, and the angle of refraction EDF.
To find these angles, subtract the reading of C from that of £,
which gives GC, and dividing by two gives EG, or CE, since
ABD equals DEG. In the same way, subtracting the reading of
C from that of F gives CF, and subtracting CE, found above,
gives FE. Dividing EG and EF by DE gives the tangents of
the angles of incidence and reflection, from which these angles
may be found. The ratio of their sines, or the difference of their
logarithmic sines, should then be constant, and give the index of
refraction. This in the case of water equals 1.33. If preferred,
AB need not equal BC, but it is more convenient to have them
both equal to some simple number, as 10 inches.
74. LAW OF REFRACTION. II.
Apparatus. The instrument represented in Fig. 52 may, by a
slight change, be employed to prove the law of refraction. The
graduated circle is mounted vertically, the needle D replaced by
a narrow slit, the mirror B removed, and a test-tube attached
to the index C. This test-tube is held by a strip of brass, whose
top is just on a level with the centre of the circle. If, then, it is
filled with water, so that the bottom of the meniscus is just above
the brass, the top of the water will be just on a level with the
centre of the circle, even if the tube is inclined. To mark the
direction of the ray in the liquid, two diaphragms with slits in
them are placed in the tube, one at the bottom, the other in the
middle. A piece of white paper, or a mirror, should be placed
below to reflect light up through the tube, and the whole should
be mounted on levelling-screws, and placed in a good light oppo-
site the window.
Experiment. By the following method, the law of refraction,
that the ratio of the sines of the angles of incidence and refraction
is a constant, may be proved more directly than in the last experi-
ment. Set the index C at 90°, so that the test-tube shall be
vertical, and move the other index carrying the slit, to 270°.
The three slits will now be in the same vertical line, and on look-
ing through the upper one, light will be seen through the other
two. If not, they must be brought into this position by moving
the test-tube. Fill the latter with water until light is just visible
INDEX OF REFRACTION. 147
above the brass strip. If now the test-tube is inclined by moving
its index, the other index must be moved by a larger amount to
bring the three slits again apparently in line, owing to the refrac-
tion at the surface of the water. And the angles through which
these indices have been moved will equal the angles of refraction
and incidence, respectively. Before making the measurement,
however, the line connecting the indices in their first position
must be brought at right angles to the surface of the water. For
this purpose turn the upper index 70° or 80°, or as far as readings
can be conveniently taken, and turn the lower index until light
passes through the three slits. Read its position and turn each
index as much on the other side. If light is again visible through
the slits, no further correction is necessary. If not, turn the level-
ling screws through one half the distance required to bring them
apparently in line. By repeating this correction the adjustment
may be made exact. Then take a number of readings of the two
indices, moving the upper one a few degrees, and turning the
lower one until light is visible through the slits. Subtracting from
these readings 90° and 270°, gives the angles of incidence and
refraction. The difference of the logarithm of their natural
sines will equal the logarithm of the index of refraction.
75. INDEX OF REFRACTION.
Apparatus. The instrument devised by Wollaston to measure
the index of refraction is represented in Fig. 58. A cube or right-
angled prism of glass A, rests on a plate of glass in which a slight
depression has been ground. The system of jointed bars JBC, CE
and DF, is attached to this, so that when C is raised, F and B slide
towards E. BC is exactly 10 inches long, and carries two sights
through which A maybe viewed. EC equals 10 inches multiplied
by the index of refraction of the prism, and DC =. DE — DF\
hence E is always vertically under (?, because if a circle is de-
scribed with centre D and radius D (7, CFE will be a right-angle,
being inscribed in a semicircle. EF is divided into inches, but
the graduation need extend only from about 12 to 15| inches, if
the index of refraction is 1.55. Bottles containing several liquids
to be measured are also needed, as water, alcohol, turpentine and
various oils, and some solids with polished surfaces, as mica, quartz,
and marble.
Experiment. Place a drop of water in the hollow under the
prism and raise G. On looking through the sights the spot where
148 CHEMICAL SPECTROSCOPE.
the prism rests on the drop is at first bright, but after passing a
certain position, becomes dark. The bounding line is marked by
colors, and bringing it into the mid-
dle of the field of view and reading
F should give 13.35, which divided
by 10, or 1.335, is the index of re-^
fraction of water. The explanation
is, that when C is low, total reflection
takes place, and the spot appears bright ; but when raised the
light is mostly transmitted. The colored line appears at the
angle of total reflection, in which case the sine of the angle
of incidence equals the ratio of the indices of refraction of
the two media. Call n and nf the indices of the glass and
given liquid, i the angle of incidence of the ray through the sights
upon the prism, r its angle of refraction, and 90° - - r its angle of
incidence on the reflecting surface of glass and liquid. Then be-
ing at the angle of total reflection, sin (90° — r) = cos r = — >
orn' = n cos r, and the problem is to prove that this equals EF,
calling CB, or 10 inches, equal to unity. Now sin CBE : sin CEB
= CE : GB= n : 1. But CBE — i, hence GEB — r. Again,
since F\% vertically under (?, EF = CE cos CEF = n cos r, or
equals nr.
Wipe the drop of water carefully from under A^ and replace it
by other liquids in turn. Next measure the indices of one or
more solids, by cementing them to the glass by a drop of some
liquid of higher index, as balsam of tolu.
76. CHEMICAL SPECTROSCOPE.
Apparatus. A common chemical spectroscope with one prism,
and a photographed scale to measure the position of the lines. Two
Bunsen burners, an Argand burner, some platinum wires sealed
into the ends of glass tubes, and two stands to hold them a little
lower than the slit of the spectroscope. A dozen small vials are
set in a stand formed by boring holes in a block of wood, and filled
with the substances to be tested. Part contain salts of sodium, lith-
ium, strontium, calcium, barium, thallium, etc., and the remainder
labelled A, B, C, etc., contain mixtures of these substances. In
Fig. 59, the light enters the instrument through a slit in the end
of the tube B. At the other end of this tube is a lens, whose
CHEMICAL SPECTROSCOPE. 149
focus equals the length of the tube, so that the rays emerging from
it are parallel, that is, the same effect is produced as if the slit
were placed at an infinite distance. The width of the slit is varied
by means of a screw acting against a spring, so that more or less
light may be admitted. The rays next encounter the prism Z>, by
which they are refracted, the different colors being bent unequally,
and then enter the observing telescope A. A series of images of
the slit are thus produced, one for each color, forming a continuous
band of colored light, red at one end, and violet at the other. To
measure the position of the different parts of this spectrum, a
third tube, (7, is employed, which carries at its outer end a fine
scale photographed on glass, and the rays from it are rendered
parallel by a lens, as in the case of J2. G is set at such an angle
that the image of the scale reflected in the face of the prism is
visible through the observing telescope A, at the same time as the
spectrum. To exclude the stray light, D must either be enclosed
in a box, or covered with a black cloth.
Experiment. Turn B towards the window, or better, reflect a
ray of sunlight through it, and nearly close the slit. A brilliant
band of color becomes visible through A,
red and yellow at one end, and blue and
violet at the other. Slide the eye-piece
of A in or out, until the edges are sharply
marked, when fine lines will be seen at
right angles to its length. These must
be focussed with care, noticing that with
a wide slit they disappear entirely, while with a very narrow one
they are obscured by other lines at right angles to them, due to
irregularities in the slit, or dust on its edges. Light the Argand
burner and place it near (?, drawing the scale in or out, until its
image is distinctly visible through A. This adjustment is aided
by closing the slit, or covering it up. Both the lines and scale
should no\v be distinctly visible, and the position of the former
may be accurately determined by the latter. Record in this way
the position of a number of the more prominent lines in. the solar
spectrum.
Tarn the slit away from the window, so that the field of view
shall be dark, and light one of the Bunsen burners, placing it
opposite the slit, and three or four inches distant. Heat one of
the platinum wires in it, until it ceases to color the flame. Then
dip it in the vial containing soda, and place it on its stand in the
150
CHEMICAL SPECTROSCOPE.
flame. On looking through A, a brilliant yellow line is visible,
which, with a more powerful instrument, is seen to be double, or
to consist of two fine lines very near together. This line is very
characteristic, and by it an almost infinitesimal amount of soda
may be detected. In fact, it shows itself in all ordinary sub-
stances. Record the position of this line, burn the wire • clean,
and repeat with lithia and the other substances. Most of them
give several lines, which sometimes become more visible after the
wire has remained in the flame for some time. Thus strontia
gives a blue line, a strongly marked orange line, and six red lines,
all of whose positions should be accurately recorded. To save
time, it is best to use two wires, observing with one, while the
other is being cleaned by heating it with the second Bunsen
burner. It may sometimes be cleaned more quickly by heating it
to redness, and dipping it in cold water, or by crushing the bead
formed with a pair of flat-nosed pliers ; but care must then be
taken not to break the wire. If the salt will not adhere to the
wire, the latter may be moistened with distilled water, or a loop
made in its end. Having measured, and become familiar with
these spectra^ try some of the contents of vial A. See first if its
ingredients can be recognized from its spectrum by the eye, and
then measure all the lines, and compare with the measurements
previously taken. The substances present may be found by the
law that the spectrum of a mixture of any bodies contains all the
lines they give separately.
All measurements made with a spectroscope must be reduced to
some common scale, in order that they may be of real value, as
the scales attached to these instruments are quite arbitrary, no
two being alike. To make such a reduction, the lines measured in
the solar spectrum must first be identified. The table given in
Expeiiment No. 77, on p. 152, may be employed for this purpose.
If the light used is only that of the sky, instead of sunlight, the
visible spectrum will only extend from near B to G. Having
identified the intermediate lines, construct points with ordinates
equal to their observed position, and abscissas equal to their wave-
lengths. A curve is thus obtained, from which any readings of
the spectroscope may be reduced to wave-lengths by inspection.
Apply it to the lines of the metals measured above.
SOLAR SPECTROSCOPE. 151
77. SOLAR SPECTROSCOPE.
Apparatus. The optical circle, a 60° prism of dense flint glass,
and a mirror capable of turning horizontally or vertically, by
which a ray of sunlight may be reflected in any desired direction.
This is accomplished more perfectly by a heliostat, in which the
apparent motion of the sun is corrected by moving the mirror by
clockwork. The instrument should be placed near a window into
which the sun is shining, or if the day is cloudy, the experiment
may first be performed with a Bunsen burner and a wire on
which is a little borax, and afterwards concluded by the aid of
sunlight.
Experiment. Adjust the optical circle as 'described in Experi-
ment 72, so that both telescopes shall be focussed for parallel rays,
and when placed opposite each other the cross-hairs and slit shall
be distinctly visible. Bring them to coincide, and read the ver-
nier. Turn the observing telescope about 45°, and place the prism
on the centre plate, so that its back shall be equally inclined to
the axes of the telescope and collimator, as at D, Fig. 59. Place
the mirror in the sunlight, and turn the collimator towards it, and
distant only a few inches. Nearly close the slit, and reflect the
light through it by turning the mirror. On moving the telescope
to one side or the other, if necessary, a brilliant spectrum will be
visible, any part of which may be brought to coincide with the
cross-hairs, and its position determined by the vernier.
To obtain the best results, the position of the mirror must be
accurately adjusted. This may be done in two ways. Most sim-
ply by opening the slit wide, when the position of the beam of
sunlight may be seen in its passage through the object-glass of
the collimator, forming a bright spot on it. The mirror should
then be turned until this spot falls in the centre. Holding a
sheet of thin paper against the object-glass renders the spot more
visible. The slit must be nearly closed before looking through
the telescope, or the eye may be injured by the intense light. A
more accurate method of adjustment is to remove the eye-piece
and look through the tube, when an image of objects reflected in
the mirror will be faintly visible. Turn the mirror until the im-
age of the sun falls in the centre of the object-glass and the
light will then pass through the axes of both telescopes. At the
same time the prism should be placed so that it shall cover as
152
SOLAR SPECTROSCOPE.
much of the object-glass as possible. If no light is seen, even
if the slit is opened wide, probably the telescopes are not set at
the right angle. The brilliancy of the image depends on the
width of the slit, and when the latter is very narrow, the image
of the sun will widen out by diffraction. Having set the mirror
correctly, it will remain right only for a few minutes, owing to
the apparent motion of the sun, and hence must be readjusted
every little while. This may commonly be done with sufficient
accuracy without removing the eye-piece. Much trouble may be
saved by noting the point on the opposite wall where the reflected
beam falls, and resetting the mirror by this. Or, a small mirror
may be attached, and the direction of its reflected beam noticed.
If the shadow of a window-sash falls on the mirror, move the lat-
ter across it, so that the further motion of the sun may separate
them instead of again bringing them together.
Next bring the prism to the minimum of deviation, that is, so
that its back shall be equally inclined to both telescopes. Turn
the prism while looking through the telescope, and the spectrum
will ]be seen to move a certain distance toward the red end, and
then return. As a considerable motion of the prism corresponds
to but a slight motion of spectrum, this point may be found with
sufficient accuracy by the hand alone. Now focussing the teles-
cope with care, the spectrum will be seen to be traversed by a mul-
titude of fine vertical lines known as Fraunhofer's lines. Bring
one of these to coincide with the cross-
hairs after setting the prism at the min-
imum of deviation, and read the vernier.
It will be found that the minimum for
one line is not the minimum for another.
Measure in the same way the position
of several of the more prominent lines ;
which may then be identified by the
accompanying table. The first column
gives the names, the second their wave-
lengths, and the third their position on
the map of Kirchhoff, which is still much
used as a standard. The line A is about the extreme limit of the
red end of the spectrum, and can only be seen in strong sunlight.
Name.
W.L.
K.
A
7605
405
a
7185
500
B
6867
594
C
6562
694
a
6276
810
D
5892
1005
E
5269
1528
h
5183
1634
F
4861
2080
G
4307
2855
h
4101
3364
HI
3968
3779
SOLAR SPECTROSCOPE. 153
JB is therefore often mistaken for it. C is a sharply marked, but
fine line in the red, caused by hydrogen, a is due to aqueous vapor
in the air, and is most conspicuous about sunset. It then bears a
marked resemblance to B. J) is a double line in the yellow, due to
soda. The fine lines between its two components were often used
as tests, until it was shown by Prof. Cooke that most of them were
due to aqueous vapor. E is a close double line in the green in the
midst of a group of double lines, some of them very close, b
consists of four very strongly marked lines, three of them due to
magnesium, of which the least refrangible is b±. They contain
several fine lines, which form good tests of the power of a spectro-
scope. F is a strong line in the blue, like (7, due to hydrogen.
G lies in the midst of a group, among a multitude of lines, h is
fine and due to hydrogen, and H consists of two very broad lines,
almost at the limit of the visible spectrum.
To determine the indices of refraction for these lines, subtract
from the reading of the vernier in each case the reading when the
telescopes were opposite, and the difference D gives the deviation.
If i is the angle of incidence, and r the angle of refraction for the
first surface, since the prism is at the minimum of deviation, the
angle of incidence at the second surface will equal r, and the an-
gle of refraction i, as both faces are equally inclined to the light.
Again, it maybe proved by Geometry, that calling A the angle
of the prism, A = 2r. The ray is deviated at each surface i — r,
hence the total deviation D = '2(i — r) = 2i — A, or i =
D). If the index of refraction equals n^ sin i = n sin r
sin \(A 4- D)
or n = - — -. . j - -. Compute in this way the index of re-
sin -f)^cj,
fraction n for each of the lines, and see if they satisfy the theoret-
B C
ical formula of Cauchy, n = A -\ — ^ -\ — -± = A -\- Bx -|- Cx\
A A
calling x equal to the. reciprocal of the square of the wave-length,
and A., B and (7, constants depending on the particular material
of which the prism is composed. By the method of least squares,
p. 4, the most probable values of A, B and C, may be found, and
compared with observation by a residual curve. To insure accu-
racy, it is safer to remeasure the indices again, using the other side
of the graduated circle and employing the mean. By using a
154 SOLAR SPECTROSCOPE.
hollow prism bounded by two plates of glass, the indices of li-
quids may be measured, and with a prism of quartz the relation
of the ordinary and extraordinary indices to the wave-length, es-
tablished.
To obtain really valuable results in this experiment, great care
is necessary, and an instrument of the finest construction. The
more powerful spectroscopes contain a number of prisms, thus
giving a much longer spectrum. In some the light passes twice
through each prism, the collimator being placed immediately over
the observing telescope, or better, united with it. With such an
instrument a vast number of lines may be seen and identified by
comparison with the maps of Angstrom or Kirchhoff. To meas-
ure the exact place of those near together, it is better to determine
accurately the position of two or three, measure the rest by a
spider-line micrometer, and then reduce to wave-lengths by inter-
polation.
The distance between the two components of a double line may
also be readily determined by the same instrument. It consists of
an eye-piece, in which are two vertical spider lines, one fixed, the
other movable by a micrometer screw the number of whose turns
is commonly measured by notches in the upper part of the field
of view, and the fraction of a turn by a circle divided into one
hundred parts, attached to the screw-head. Both wires may be
moved by a second screw, and illuminated by a light placed oppo-
site a piece of glass inserted in one side. It is used when the field
is dark, to render the lines visible. The distance between two
lines may be measured as follows. Call the screw with divided
head, A, and the other, B. Bring the two cross-hairs to coincide,
and read the micrometer-screw, A, repeat several times, and
take the mean. Then turn B until the fixed hair coincides with
one line, and turn B until the movable hair coincides with the
other. The reading of A minus that previously taken, gives
their distance apart. After setting both hairs, their position is
sometimes reversed, and the distance through which A has been
turned equals double the distance between the lines. It is a good
exercise to measure all the lines visible in a small portion of the
spectrum, and then compare with one of the charts mentioned
above.
LAW OF LENSES. 155
If the sun is not shining, the Bunsen burner may be employed
instead, using the platinum wire with a borax bead at its end.
This will give a bright, double line, coinciding exactly with the
dark line, D, in the solar spectrum. Its position, and the interval
between its components, should be accurately determined. Spec-
tra of great beauty may also be obtained with an induction coil
by allowing the spark to pass between terminals of different met-
als placed in front of the slit. Still finer effects are obtained with
the electric light.
78. LAW OF LENSES.
Apparatus. In Fig. 60, A is a fishtail burner, attached to the
end of a bar eleven feet long, and divided into tenths of an inch.
B is a lens of two feet focus, by which an image of A may be
projected on the screen O. Both B and C are movable, and carry
pointers to show their distance from A.
Experiment. Place C at the end of the bar, and B just 100
inches from A. An image of the flame will be formed on (7,
which is then moved backwards or
•^0 A & forwards until the exact focus is
A T found. When the screen is too
.,..,....^..,...,.-^.-..-. ..,....,.. .(.. »..J| near, it will be noticed that owing
to chromatic aberration, the edges
of the image are red, while if too
distant they are blue. The intermediate position may thus be
found with great accuracy. Read and record the distance A C, and
repeat, making AB successively 95, 90, 85, etc., inches. C will
approach A up to a certain point, until AB — J3C equals twice
the focal length of the lens. Determine this point more exactly
by taking a number of readings, moving B an inch at a time.
Then continue to diminish AB two inches at a time, until the
image falls off the bar.
Write the results in a table in which the first column contains
AB, the second AC, and the third their difference, or BC. Now
compute the true value of BC in each case, and insert in the
fourth column. Calling u and v the conjugate foci, or AB and
156 MICROSCOPE.
BC, and /the principal focus of the lens, - -j — = - = -j-= -{-
- In this formula AJBis successively made equal to 100, 95, etc.,
inches, and /equals one fourth the minimum value of AC. By a
table of reciprocals the calculation is easily made by subtracting
the reciprocal of AB from the reciprocal of/, and the reciprocal
of the difference gives BC. Construct two curves on the same
sheet of paper with the same abscissas AB, a&d ordinates equal to
the observed and computed values of B (7, respectively, and their
agreement proves the correctness of the formula.
79. MICROSCOPE.
Apparatus. The importance of this instrument renders it de-
sirable that each student should devote considerable time to its
use. For this reason, in a large laboratory two or three micro-
scopes should be procured, and it is well to have them from differ-
ent makers, so that the student may be accustomed to all forms.
For example, a "Student's Microscope," by Tolles or Zentmayer,
to represent the American instrument, a binocular " Popular
Microscope," by Beck, for the English, and a third instrument by
Nachet or Hartnack, for the Continental form. The latter is very
cheap and good, but not having the Microscropical Society's screw,
common objectives cannot be used on it without an adapter. It
is also well, if it can be afforded, to have one first-class microscope-
stand for work of a higher nature. The usual appurtenances de-
scribed below should be added, but need not be duplicated, also a
number of objectives and objects.
The following description will serve for all the common forms of
instrument. A brass tube or body is attached to a heavy stand,
so that it can be set at an angle, or moved up or down. In its
lower end the objectives are screwed, and the -lye-pieces slide into
the upper end. The objectives are made of three achromatic
lenses, by which a short focus is attained, with great freedom from
aberration. The eye-piece is of the form known as the negative
eye-piece, and consists of two plano-convex lenses, with their
plane surfaces turned upwards. Below is placed the stage, on
which the object is laid and kept in place, either by a ledge, or by
spring clips. In the larger stands the object may be moved by
two racks and pinions in directions at right angles, or revolved by
turning the stage. It is very desirable that this rotation should
take place around the axis of the instrument, as is done in the Eng-
lish, but not in the American instrument mentioned above. Un-
der the stage is the diaphragm, a plate of brass with a number of
MICROSCOPE. 157
circular holes in it of different sizes, to admit light more or less
obliquely. Below it is a mirror, plane on one side, and concave on
the other, by which light may be reflected upon the object.
It is very important that the body of the instrument may be
raised and lowered with precision. There are generally two ad-
justments to effect this, one the coarse adjustment to move it rap-
idly, which is commonly a rack and pinion, or a simple sliding
motion effected by hand, and a fine adjustment which is used for
getting the exact focus, and is made in a vaiiety of ways. One of
the best is by a movable nose-piece, or the lower end of the tube
made free to slide, and acted on by a lever, which may be moved
by a screw. In a second form, the screw acts directly on a part of
the tube itself, and sometimes the stage is raised or lowered. If the
tube is moved, it should be raised only by the screw, the lowering
being effected by a spring, so as to prevent the objective from
being pressed forcibly against the object.
To show the use of each instrument used in connection with
the microscope, one or more objects should be selected suited to
each, and numbered as in the foltowing examples. They may
then be distributed among the various microscopes, according to
the means or requirements of each laboratory. In this way a
student acquires a better knowledge of the apparatus, and of the
proper objects to which each appliance is best suited, than he
could attain in weeks of unsystematic work. It is also well to
examine several of the objects described below, with various
methods of illumination, to learn how much their appearance may
be thus altered. When studying a new object it should always be
illuminated in various ways, and viewed first with a low, and after-
wards with a higher power. There is, however, no more common
mistake than to suppose that objects will be seen better, the higher
the power. On account of the difficulty with which they are
used, the want of distinctness and of sufficient illumination, the
highest magnifying powers must be reserved for special occasions,
and the lower powers commonly employed, especially in the pre-
liminary observation of common objects. To save the eyes, it is
better, at first, not to use a microscope very long at a time, and for
the same reason, they should both be kept open. If possible,
sometimes one eye should be used, and sometimes the other.
The applications of the microscope have been so extensive that
it is impossible in a short article like the present, to give more
than a general description of the most important. The student
who wishes to make, a specialty of this instrument is therefore
referred at once to some of the works devoted especially to this
subject. For instance, the treatises of Carpenter, Hogg and Beale,
particularly the work by the latter author, entitled " How to Work
with the Microscope." The same remarks apply with even more
force to Experiments 80 and 81.
158 MICROSCOPE.
Experiment. 1. Ordinary Method of using the Microscope.
Set the microscope in an inclined position, at such an angle that it
can be used with comfort. The tube carries an objective below,
by means of which an enlarged image of the object is formed, and
magnified a second time by the eye-piece. Slip into the upper
end of the tube the lowest power eye-piece, that is, the longest.
The objectives are contained in brass, cylindrical boxes, with screw
covers. They must be handled with care, as they are very expen-
sive; the higher powers consist of very minute lenses, and the
glass surfaces must never be touched, lest they be tarnished or
scratched; the lower surface, which is plane, is particularly ex-
posed to injury. Remove the cover of an objective whose focus is
one or two inches, and screw it into the tube. Now turn the mir-
ror so that the light from the window shall be reflected along the
axis of the instrument, and on. looking in, a bright circle of light
will be visible.
Place object No. 1, eye of a fly, on the stage, and raise or lower
the tube until it is distinctly visible. The distance between the
objective and object should be somewhat less than the focal length
of the former. Notice that the eye is composed of a multitude
of facets, like the meshes of a net, each one containing a separate
lens. Sketch some of them in your note book. Try the other
eye-piece, which will give a somewhat higher power. Then re-
move the objective, putting it back in its box, and replace it by
one whose focal length is £ inch. The use of this is attended with
somewhat greater difficulty. It must be brought very near the
object, but not in contact, or it would very likely be scratched, or
even broken. It is therefore safest to bring it as near as possible
without touching, by the coarse adjustment, then looking through
the instrument to withdraw it until the distance is about right, and
finally focus exactly, by the fine adjustment. A great increase of
magnifying power is thus attained ; add to the description and
drawing whatever additional is visible. Do the same with a sec-
ond object, foot of the Dytiscus.
2. Diaphragm. Immediately under the stage is a brass plate
pierced with a number of holes of different sizes. Its object is to
vary the amount of light and the direction in which it comes.
When a small aperture is used, all the light comes in nearly the
MICROSCOPE. 159
same direction, and thus renders the shadows of minute objects
more distinct. The structure of delicate objects is thus some-
times brought out very beautifully, where a large aperture con-
ceals everything. To show this, try object No. 2, proboscis of a
horse-fly, and see how much more distinct the fine hairs at the end
are, with the small aperture. Also the diatom Jsthmia nervosa, in
which the markings, although perfectly distinct with a small
aperture, almost disappear when the diaphragm is turned so as to
admit a large cone of light.
3. Oblique Illumination. Microscope objectives are made so
that they will transmit rays of light not only along their axis, but
also when falling obliquely on them, that is, they will receive a
cone, the angle of whose vertex is called the angular aperture of
the objective. For the higher powers this angle is sometimes very
great, 170°, 175°, or even 177°. With them, instead of placing
the mirror immediately underneath, it may be placed on one side,
and the object illuminated obliquely. A better plan is by an
Amici's prism which is placed below the object, and throws the
rays obliquely like the mirror. The advantage in this case is like
that of a diaphragm, only greater, shadows being strongly cast, and
very delicate structure rendered visible. This effect is well shown
with many diatoms, minute siliceous shells, on which are markings
or very fine parallel lines, used frequently as tests. Try the quarter
inch objective on specimens No. 3, Pleurosigma formosum and
Pleurosigma hippocampus. First use direct light and then an
oblique illumination, and see how much more distinctly the mark-
ings are visible in the second case.
4. Opaque Objects. Some objects, especially those of large size,
cannot be rendered transparent, and sometimes the surface only of
a body is to be examined. In this case remove the mirror from
below its object, and place it above on a stand, turning it so that
the light shall be thrown down upon the object. A second method
is to use for the same purpose a large lens of short focus called a
condenser. If the observer is facing the window, it is generally
necessary in this case, to place the object nearly horizontal, in
order to get light upon it. Try both these methods on objects No.
4, wing of a butterfly, and section of bone or tooth, viewing the
latter also by transmitted light.
160 MICROSCOPE.
5. Lieberlvuhn. Another method of illuminating opaque ob-
jects is by a parabolic mirror, with a hole in its centre, through
which the objective is passed. This device, called a lieberkuhn, is
used on small opaque objects, the light being thrown from the
mirror below upon the lieberkuhn, and by it reflected upon the
object. Try specimen, No. 4, wing of butterfly, thus illuminated,
also some common objects, as a bit of paper, a steel scale, etc.
6. ~Wenhairts Parabolic Condenser. This consists of a block
of glass, plane below and parabolic above. It is placed, instead
of the diaphragm, just below the object, which is at its focus, so
that all light reflected upon it by the mirror below, will fall on the
object illuminating it obliquely. The central rays are cut off by a
circle of metal attached to the condenser. Objects are thus shown
bright on a dark background, sometimes producing an excellent
effect, though generally more beautiful than useful. See No. 6,
Arachnoidiscus Ehreribergii.
7. Achromatic Condenser. The mirror below the object is
commonly plane on one side, and concave on the other, the former
reflecting light on a given point from various directions, the latter
concentrating that received from a single point. The second form
is more commonly used, especially with artificial light, as any
point may thus be selected as the source of illumination. The
same effect is much better attained by placing below the object an
objective similar to that above it, which allows only those rays
parallel to its axis to pass through both. As it costs too much to
duplicate all the objectives, each may be used as an achromatic
condenser to that of next lowest power. This is a very favorite
method of illumination, especially when using high powers on dif-
ficult objects. Try it on No. 7, fragment of hair and Surirella
gemma.
8. Polariscope. One other method of illumination remains to
be described; namely, that by polarized light. To use this to the
greatest advantage, Experiment 88 should first be performed. The
light is polarized by a Nicol's prism, placed under the object to be
examined instead of the diaphragm, and a second prism or analyzer
is placed above it, either slipping it over the eye-piece, or screwing
it onto, and just above, the objective. On rotating either analyzer
or polarizer, the field becomes dark when their planes are at right
MICROSCOPE.
161
angles, nnd brightest when they are parallel. Sometimes a plate
ofselenite is inserted just above the polarizer, when the field will
assume a brilliant color, which may be changed into, its comple-
mentary tint, by revolving one of the Nicol's prisms. To examine
any object by polarized light, focus it as usual, and turn one of the
prisms ; the most marked effect is generally attained when they
are at right angles, as with common objects, the field would be
perfectly black, while any doubly refracting medium will appear
bright in places, bringing out the structure with great beauty.
Now insert the plate of selenite, and the uniform tint will, in
many cases, be replaced by a gorgeous display of colors. Examine
the following objects, No. 8, section of cuttle-fish bone, quill, sul-
phate of magnesia, nitre and salicine. In fact, any crystalline sub-
stances not in the monometric system, affect polarized light, and
the same may be said of many organic structures, as starch, bone,
hair and horn.
9. Binocular Microscope. To avoid the fatigue of using one
eye only, and to obtain the stereoscopic effect due to two, this
instrument is employed. A small prism is placed just above the
objective, which divides the light into two portions, which pass
along two tubes, one for each eye. Two eye-pieces are of course
used, and on looking through them, the object is seen as in the
stereoscope, standing out in its true form. The distance between
the eye-pieces may be altered by lengthening or shortening their
tubes by a pinion acting on two racks, and the microscope may be
rendered monocular by merely pushing back the binocular prism.
This instrument is best suited to opaque objects not requiring a
high power. Apply it to objects No. 9, head of a bee, and an
injected preparation of the upper portion
of the lung of a frog, using the 1 or 2 inch
objectives and the lieberkuhn.
10. Maltwood's Finder. Some objects
are so minute that they are found only
with difficulty, and it is also sometimes
desirable to refer back to a certain point
of an object and reexamine it. A Malt-
wood's finder consists of a photograph on
glass, one inch square, of a series of squares numbered as in Fig.
11
30
31
32
51
51
51
30
31
32
52
52
52
30
31
32
52
52
52
162
MICROSCOPE.
61, all the lower numbers in the same horizontal row being the
same, also all the upper numbers in the same \ertical column.
In other words, the upper numbers give the abscissa, the lower
the ordinate, of each square. This photograph is then mounted
on a slip of glass like any other object. A pin or stop should be
attached to the stage, so that both finder and object may al-
ways be placed in precisely the same position. Lay the object
on the stage and bring the point to be recorded exactly in the
centre of the field, using a moderately low power, if the object
is not too minute. Now put the finder in the place of the
object, taking care not to move the stage, and record the numbers
of the square in the centre of the field. If at any time the same
point of the object is again wanted, the finder is placed on the
stage, and the latter moved until the square bearing these numbers
is again in the centre of the field. Replacing it by the object, the
desired point should be at once visible. Apply this method to
objects No. 10. First, with the preparation of an entire insect, as
a gnat, record the numbers corresponding to several prominent
points, as the eye and the end of the proboscis. Then move the
stage and see if they can be found again. Do the same with a
slide containing a single minute object as No. 7, Surirella gemma.
Now, placing on the stage an object containing a collection of
diatoms, select a good specimen, sketch its position and that of the
adjacent ones, and see if it can be found again from its numbers.
The numbers corresponding to the marked points of some of these
objects should be recorded on a card placed with the microscope,
and these points then found by the student.
11. Micrometers. There are several forms of this instrument,
which is used for measuring minute objects. First, the stage-mi-
crometer, which consists of a plate of glass ruled with fine lines at
equal intervals of T^W m- °r TOTT mm' The lines are so delicate
that it is often difficult to find them, as a slight difference of focus
throws them out of sight. Their visibility may be increased by
oblique illumination, or by using a small aperture in the diaphragm.
The eye-piece micrometer consists of a coarser scale on glass, in-
serted at the focus of the eye-piece, which, in the negative form, lies
between the lenses. An enlarged image of it is thus formed in the
field, which is used like an ordinary scale to measure the dimensions
MICROSCOPE. 163
of objects. To reduce them to fractions of a millimetre, lay a stage
micrometer on the stage and measure its divisions. For instance,
if 7 hundredths of a millimetre equal 53.4 divisions of the microm-
eter, one division of the latter will equal .00131 mm. From this
any measurement made with the micrometer may be reduced to
millimetres. If the microscope has a draw-tube, this reduction may
be simplified. In this case, the tube of the microscope is made
double, so that its length may be altered, a scale showing the
extent of the change. The power is increased by drawing out the
tube, and the divisions of the stage-micrometer being enlarged,
they may be made exactly equal to any desired number of divi-
sions of the eye-piece micrometer. Thus, in the above example,
make the 7 divisions equal to 56, instead of 53.4, when one will equal
8, or each division of the eye-piece micrometer equal one eight
hundredth of a millimetre. In the same way make it one thou-
sandth, and altering the objective, give it other values, recording
in each case the reading of the draw-tube.
Now make a number of measurements of objects No. 11, as
directed on a card, which should accompany the specimens, and
reduce them to millimetres. Determine also the thfckness of a
hair, of a filament of silk, and the diameter of some minute holes
made in a piece of tin-foil with a fine cambric needle. In the
spider-line micrometer, the two hairs are brought to coincide with
the ends of the distance to be measured, and the interval deter-
mined, as described in Experiment 77. The readings should be
reduced to fractions of a millimetre, in the same way as the eye-
piece micrometer. Repeat the above measurements, and see if the
same results are obtained as before. The magnitude of the divi-
sions of both the spider-line and eye-piece micrometers, depends
only on the distance from the objective and its focus, and not at
all on the eye-piece, unless a negative eye-piece is used, and the
micrometer inserted between the field- and eye-lenses.
12. Goniometers. A spider-web, or filament of silk, is stretched
in the eye-piece across the field of view, and turned so as to coin-
cide first with one side, and then with the other, of the angle to be
measured. The number of degrees through which it is moved
gives the required angle. A graduated circle is therefore attached
to the tube, and an index is fastened to the eye-piece. Sometimes
164 MICROSCOPE.
the object is moved instead, the graduated circle being attached to
the stage. To test this instrument, form a triangle by selecting
three points on any object, and suppose them connected by
right lines. Measure each angle two or three times, displacing the
object and eye-piece after each measurement ; take the mean, and
see if the sum of the three equals 180°. Now measure several
angles of the crystals, No. 12, chlorate of potash, taking care that
they lie flat, otherwise too small a value of the angle will be
obtained.
13. Camera Lucida. It is often important, when studying
minute objects by the aid of the microscope, to be able to draw
them correctly. For this purpose, the enlarged image must be
thrown on the paper in such a way that both may be distinctly
seen at the same time. This is done most simply by keeping both
eyes open, and directing one towards the paper the other through
the microscope, when the image and paper, may be brought to-
gether so that the outlines of the former may be marked on the
latter. A better method however, is by the camera lucida, which
consists of a minute right-angled prism of glass, having its acute
angles equal to 45°. Place the microscope horizontally, adjust
the mirror so that the field shall be bright, and apply a low power
to object No. 13, a flea. Attach the camera to the eye-piece so
that on looking down into the mirror, a reflection of the object
shall be thrown on a sheet of paper, placed immediately beneath,
in which case one face of the prism will be horizontal and turned
upwards, the other vertical and turned towards the microscope.
The prism is of small size, so that it will cover only a part of the
pupil of the eye, and bringing the latter over its edge, both paper
and object may be seen simultaneously. With practice, the out-
line may be marked out very accurately, but at first it is diffi-
cult to see the pencil at the same time as the object. As the latter
is often too bright, it is sometimes better to incline the mirror un-
til the field is darker. Better results are also sometimes obtained
if the eye is raised an inch or so above the prism.
As with this instrument, the brilliancy of the object generally
renders it difficult to distinguish the pencil, the prism is sometimes
replaced by a small piece of plate glass, which reflects the image
in the same way as the back of the prism, while its transparency
MICROSCOPE. 165
renders the pencil visible through it. The latter is in fact gen-
erally too bright, so that it is often better either to cast a shadow
on the paper, or to make the glass of a neutral tint, to render it
less transparent. Try making drawings with each of these instru-
ments, first of some well marked object as the flea, and afterwards
of some fainter object as the anchor-like spines of the Synapta.
Generally only the outlines should be drawn with the camera, and
the details filled in by the eye. After drawing the object, replace
it by a stage-micrometer, and draw some of the divisions of the
latter, which thus serve as a scale, by which the magnitude of
different parts of the object maybe determined. This also furnishes
the best means of measuring the magnifying power, dividing the
dimensions of the scale as drawn, by its real size. This will be
correct, however, only when the distance from the camera to the
paper is just 10 inches. In other cases it must be divided by the
distance in inches, and multiplied by 10, to reduce it to the stand-
dard. Make this measurement with the 1 inch and the £ inch
objectives, and two of the eye-pieces, also if there is a draw-tube,
make the magnifying power some simple number, by varying the
distance between the objective and eye-piece.
14. Spectrum Microscope. This consists of a spectroscope in-
serted in the eye-piece of a microscope in such a way, that the
spectrum of very minute objects may be obtained. The form in
most common use is that devised by Sorby. The slit replaces the
diaphragm, and is partly covered by a right-angled prism, by which
a second spectrum, from a light at one side of the eye-piece, may be
compared with the other. The prisms are placed over the eye-lens,
and are of the form known as direct vision, in which the deviation
of two prisms of heavy flint glass is compensated by that of three
crown-glass prisms, while the dispersion is only partly neutralized.
Accordingly a spectrum of considerable length is obtained, while
there is no deviation of the central portion. The width of the
slit may be varied by.a screw, and its length by a sliding stop. An
ingenious scale is provided, formed of two Nicol's prisms, with a
plate of quartz between them, and placed in the path of the rays
reflected by the right-angled prism. They absorb from the visible
spectrum twelve black bands, at regular intervals, and from their
position, that of any line may be readily determined. To use this
166 MICROSCOPE.
instrument, slip it on the end of the microscope in the place of the
eye-piece, and place object No. 14, human blood, on the stage with
a one or two inch objective. The spectrum will now be seen to be
traversed by two marked black lines in the red, which form an
excellent test for the presence of blood. Their position may be
measured with the scale, by attaching the latter to the side of the
eye-piece, and adjusting the prism so that the spectrum for one
half its breadth shall be traversed by strongly marked black bands.
Other objects, such as nitrate of didymium, permanganate of pot-
ash and aniline violet, may be observed in a similar manner. Care
should be taken to make all the light pass through the object,
which is generally best accomplished by placing a cardboard
diaphragm with a small hole in it, on the stage under the object.
Liquids are placed in glass tubes or cells, which may be closed
hermetically.
15. Test- Objects. The principal eiforts of microscope makers
are now directed towards the objectives, since it is by perfecting
them that the greatest improvements are to be expected. The
best method of judging of the excellence of an objective, or of
comparing those of different makers, is by trying them on a num-
ber of objects called test-objects, some parts of which can be seen
only with difficulty. To obtain the best results great skill is
needed, especially in arranging the illumination, and it must not
be forgotten that some objectives give the best results with one
class of objects, others with another. For instance, some with a
large angular aperture, give fine effects with objects requiring a
very oblique illumination, but are not suited to those of considera-
ble thickness, requiring great depth of focus.
When an objective is perfectly corrected for chromatic aberra-
tion, and a plate of thin glass is interposed between it and the
object, a new correction for color becomes necessary, in amount
depending on the thickness of the glass. This is commonly
effected by varying the distance of the front lens from the other
two, which is accomplished by turning a milled head near the end
of the objective. A divided circle and index serve to mark the
position, which will of course vary with each different object, ac-
cording to the thickness of the covering glass. To make this
correction, adjust the objective for an uncovered object, that is,
PREPARATION OF OBJECTS. 167
set the index at zero and focus it on the object. Then turn the
milled head until the dust on the upper surface of the covering
glass is in focus, when the proper correction will have been ap-
plied. Focussing again on the object, the latter will be more
sharply denned than before. The correction for covering glass, as
it is called, must be applied to all objectives of higher power than
i inch, to get the best effects, especially when they have a large
angular aperture. Instead of moving the front lens, it is better to
have it fixed, and to have the other two movable, as all danger of
scratching or breaking the objective and object by bringing them
in contact, is thus avoided.
Try some of the higher power objectives with the test-objects
No. 15. One of the most common tests for denning power is the
marking of the scales of the wood-flea (Podura plumbea), which
are covered wTith delicate epithelial scales, like the tiles of a roof.
Try also the hair of the Indian bat, and of the larva of the Der-
mestes. Some of the Diatoms described above, form excellent
test-objects. The valves of the genus Pleurosigma are covered
with fine markings, which form an excellent test for separating or
penetrating power. For instance, the three species, formosum,
hippocampus and angulatum, form a series of increasing difficulty,
well adapted to test objectives of ordinary power. The marking
of the first and third are apparently covered by three series of
fine parallel lines, dividing the surface into hexagons, and of hippo-
campus by two series, forming squares, but in reality probably due
to a multitude of very minute hemispheres with which the surface
is covered. The same effect may be seen on an enlarged scale, in
a common form of book-cover. Probably the best test of this
kind is a plate of glass with very fine lines ruled on it. M. Nobert
of Griefs wold has made such plates with a series of bands formed
of lines at various intervals up to a 112,000th of an inch.
80. PREPARATION OF OBJECTS.
Apparatus. A microscope with objectives and eye-pieces, sev-
eral vials containing the substances to be examined, a number of
glass slips three inches long and an inch wide, some of which have
concave centres, that is, a concavity ground out on one side, and
some circles of very thin glass.
168 PREPARATION OF OBJECTS.
Experiment. To examine a liquid under the microscope, dip a
glass rod or tube into it, and place a small drop on one of the glass
slides. Cover it with a circle of very thin glass, which will be
held in place by capillarity, and wipe off the superfluous liquid
carefully. A concavity is commonly ground in the centre of the
slide to hold more liquid, and to keep the cover in place. Exam-
ine the following objects in this way, describe and sketcli them,
and compare their appearance with that given in the works on the
Microscope, referred to in the last experiment. A drop of vinegar
viewed with a low power, is seen to be full of eels in active motion.
Milk contains multitudes of oil globules, which when united form
butter, and organic matter whose appearance furnishes an excellent
test of its purity. Blood is a curious object under the microscope.
It is most readily obtained from the finger just below the nail.
With a quarter-inch objective, it is seen to consist of a clear liquid
or serum, in which a vast number of blood-corpuscles are floating.
These are circular disks, thicker around the edge, and interspersed
with larger white globules. In its natural state the Jblood is too
thick to be conveniently observed, the corpuscles overlap, and soon
begin to shrivel up, as the blood dries. If diluted with water
osmotic action ensues; they swell up and sometimes burst. Salt
water is therefore preferable, or better still, the serum or liquid
portion which separates from the clot when blood coagulates.
Powders are sometimes viewed dry, but generally it is better to
wet them, as they are thus rendered more transparent. Place a
very minute quantity of starch on a slide, add a drop of water,
and cover with a piece of thin glass. Viewed by polarized light,
each grain is seen to contain a black cross, which changes to white
on rotating the analyzer. This cross is characteristic of starch^
and often serves to detect its presence. It is best seen in the larger
grains, as those of potato starch, and assumes brilliant colors if a
plate of selenite is interposed. The adulterations of coffee, cocoa,
etc., are readily detected by examining them in powder under the
microscope.
It is often necessary to pick up small objects under water, or to
capture a minute animal without injuring it. A good example of
this kind is the little Cyclops, often found in great numbers in
common pond water, especially in the spring. Collecting the
PREPARATION OF OBJECTS. 169
water in a white porcelain vessel, as a large evaporating dish, a
close examination will often reveal dozens of them. Their num-
ber may also be increased by filtering a considerable quantity of
the water through a cloth, which retains them, and from which
they are easily washed into the dish. To place one on the slide,
take a small glass tube about half a foot long, close one end by
the finger, and immerse the other in the water. Bring it near the
Cyclops and suddenly remove the finger, when the water will rush
in, carrying the animal with it. Replacing the finger, the tube
may be removed, the water allowed to escape a drop at a time,
and the Cyclops finally deposited on the slide. Instead of a slide
with concave centre, it is better for so large an object as this, to
use an Animalcule-Cage. This consists of a small circle of glass,
on which a drop of water containing the object is laid, and the
cover pressed down upon it by means of a brass ring, so as to
leave a space of any desired degree of thickness. Delicate objects
are thus protected from injury by crushing. A wonderful variety
of animalcule and of fungoid growths may be found in stagnant
water, or sour flour-paste, in fact in almost any decaying animal or
vegetable matter.
Minute air-bubbles are often found in various objects. To be-
come familiar with their appearance, examine a drop of soap-suds,
or gum-water containing them. They look like black, highly pol-
ished, metallic balls, with a broad, dark outline, and bright centre.
The formation of crystals is readily watched under the micro-
scope by placing a drop of the hot saturated solution on a slide,
and allowing it to cool. Try in this way sugar, phosphate of
soda, and oxalate of ammonia, first using ordinary, and then po-
larized light.
A most instructive experiment is to watch the circulation of the
blood in the foot of a frog. The animal is first rendered insensible
by means of ether or chloroform, then put in a linen bag and well
wet with water. Draw one of the hind legs out of the bag and
tie it down upon the slide, supporting the frog on a piece of wood
or frog-plate. Tie threads to three of the toes, so as to stretch the
membranes between them, and on examining it with a half-inch
objective the blood corpuscles will readily be seen passing from
the arteries through the capillaries to the veins. By putting alco-
170 MOUNTING OBJECTS.
hol or salt on the foot, all the phenomena of inflammation and its
cure may be observed. The black spots distributed over the
membrane are due to the pigment. The circulation may also be
observed in the tail of a stickle-back, or other small fish, or of a
tadpole. The latter animal, when very small, forms a beautiful
object with a low power and binocular microscope, as it is suffi-
ciently transparent to render visible the action of the heart, and
other internal organs. The effect is also improved by keeping the
tadpole for some time previously without food.
Another interesting experiment is to watch the ciliary action,
which in many of the lower animals takes the place of the circu-
lation. Cilia consist of minute hairs, which vibrate rapidly back
and forth, and thus establish currents in the liquid in contact with
them. They may be seen by scraping a little of the mucus from
the roof of the mouth of a frog, or better, from the gills of an
oyster or mussel.
Most solid substances, like wood or bone, are best seen in thin
sections, which are made as will be described in the next Experi-
ment. Fine filaments, as silk, wool or hair, are viewed by trans-
mitted light, and generally give better effects when wet with
water or oil. Some solids, especially when highly colored, are best
seen as opaque objects, with a condenser or lieberkuhn.
81. MOUNTING OBJECTS.
Apparatus. Boxes may be obtained containing all the appara-
tus needed for mounting objects, such as glass slips, thin glass
covers, Canada balsam, gold size, a stand on which slides may be
heated, a whirling table for making cells, section-cutting appara-
tus, and other devices which will be described below.
Experiment. Objects are mounted in various ways, according
to their size, and whether they are best seen dry, or immersed in
some liquid. They are protected by a circular piece of glass,
made very thin on account of the short working distance of high-
power objectives. These circles are cut from a sheet of thin glass
with an instrument like a very small beam-compass, the point
which serves as a centre, being replaced by a flat disk, and the
pencil, by a diamond. Only a faint scratch is needed, but some
skill is required, or many of them will be broken.
MOUNTING OBJECTS. 171
Try each of the following methods of mounting objects, and if
successful, cover the slides with paper and label them, giving also
your name and the date. Unless the object is very thin, or if it
is liable to be injured by pressure, it must be protected by a cell.
This consists of a circular or square enclosure, on which the thin
glass plate is laid, so as to leave a space between it and the slide.
Cells may be made of various materials, as paper, cardboard, or
tinfoil, and fastened to the glass by gum. These are very con-
venient for mounting objects dry, especially such as are not in-
jured by the air. Mount in this way some crystals of bichromate
of potash. Shallow cells may be made of Brunswick black, ap-
plying it with a brush. They are best made in a circular form by
Shadbolt's apparatus, in which the slide is placed on a small turn-
table, which is made to revolve rapidly by drawing the forefinger
of the left hand over a milled head attached below, while the
brush is held in the right hand. If the plate is warmed, the black
will dry rapidly, and the thickness of the cell may be increased by
applying successive layers. Make several such cells for some of
the objects to be mounted in balsam, as described below. To
preserve a liquid, or an object of considerable size, thick cells
are employed, which may be procured ready-made of glass.
They may be cemented to the slide by marine glue, warming them
sufficiently to melt it, removing the superfluous glue by a sharp
knife, and washing it clean with a solution of potash. Fill such a
cell with some liquid, as vinegar, and fasten on the cover with
marine glue. Take great care thatno air bubbles enter, and that
the joints are perfectly tight.
The best method of mounting the parts of insects, sections of
wood or bone, and in fact most substances, is in Canada balsam.
The object, as the foot of a fly, must first be dried and freed from
air-bubbles. For this purpose it should be heated nearly to the
boiling point of water, or placed under a bell-glass containing
concentrated sulphuric acid. To remove the air it should be
soaked in turpentine and gently warmed ; a much more effective
method is to place the whole under the receiver of an air-pump
and exhaust. Now lay the slip of glass on a little stand of brass,
and heat it by means of a spirit-lamp, or Bunsen burner. Take up
a little Canada balsam on the end of an iron wire, and lay it on
172 MOUNTING OBJECTS.
the slide, when the heat will render it perfectly fluid. Pick up the
object on the point of a needle, immerse it in the balsam, and then
cover it with a piece of thin glass. Great care must be taken that
both slide and covering-glass are perfectly clean, and that no dust
gets into the balsam, as otherwise the object will be much dis-
figured when viewed under the microscope. The main difficulty
is to prevent air-bubbles remaining on the slide. If present, they
may be removed by a cold wire, or burst by touching them with a
hot needle. The covering-glass must be lowered into place very
slowly, or bubbles will adhere to its surface. The whole is then
put away to harden under pressure, and the superfluous balsam
afterwards removed by the aid of a little turpentine.
The structure of objects of large size is generally best seen by
cutting thin sections of them, so that they may be rendered nearly
transparent, and be viewed by transmitted light. Soft substances,
as vegetable or animal tissues, may be cut with a sharp knife or
scissors, or better, with a Valentin's knife, which has two parallel
blades whose distance apart may be varied by a screw. They
should be well wet with water or glycerine, or the section will
adhere to them.
Harder substances, as wood or horn, are cut in thin sections by
forcing them through a hole in a thick brass plate, cutting off the
projecting portion, pushing it through a little farther, and cutting
again. By means of a screw, sections of any desired thickness
may thus be obtained. Cut longitudinal and transverse sections
of a piece of pine wood, first soaking it in water, and mount them
in Canada balsam. Cut also some thin transverse sections of
hair by fastening a number of them together with gum so as to
form a solid mass ; cut a thin section, and then dissolve the gum
in water.
To cut a thin section of still harder substances, as bone, quite a
different method must be employed. A thin piece is first cut off
with a fine saw, such as is used* for cutting metals ; it is then filed
thinner, and finally ground down to the required thickness with
water between two hones. On examining the section thus obtained,
it will be found covered with scratches, which must be removed
by grinding it on a dry hone, and afterwards polishing it -on a sheet
of plate glass. Prepare two such sections, soak one in turpentine
FOCI AND APERTURE OF OBJECTIVES. 173
and remove the air by means of a pump, and then mount both in
Canada balsam. The difference in their appearance will be very
marked, the one from which the air has not been removed appear-
ing full of black spots or lacunaB, formerly called bone corpuscles.
They are really cavities filled with air, which in the second speci-
men is replaced by the turpentine.
This experiment is well supplemented by performing some
dissections of animal and vegetable substances, injecting tissues,
and mounting thin sections of them.
82. Foci AND APERTURE OF OBJECTIVES.
Apparatus. Two instruments are needed for this Expeiiment.
First, a microscope with a positive eye-piece, a spider-line or eye-
piece-micrometer, and a stage-micrometer, also several objectives
to be measured. To measure the angular aperture, a graduated
circle is employed with an arm and index, to which is attached a
short brass tube, like the body of a microscope. The objective to
be tested is screwed into one end of this tube, and a positive eye-
piece slipped into the other. The tube is made so short that when
the objective is directed towards a distant object, the image formed
may be viewed by the eye-piece. To obtain a higher magnifying
power, the eye-piece may be replaced by a compound microscope,
like that used in Experiment No. 20. To obtain an accurate
measurement when the object observed is not very distant, it is
essential that the end of the objective should lie in the axis of the
circle. This is most readily accomplished by means of a ledge, on
which a vertical plate of glass may be placed with its front face
over the axis of the circle. The objective is then brought up in
contact with it, the tube clamped, and the glass removed.
Experiment. To measure the focal length of an objective it is
assumed that two of the laws of simple lenses hold for a com-
pound lens. First, that the sum of the reciprocals of the conju-
gate foci equals the reciprocal of the principal focus, and secondly,
that the ratio of the magnitudes of the object and image equals the
ratio of the conjugate foci. This assumption is not strictly cor-
rect, and valuable work might be done in determining the amount
of the deviation. Screw the objective to be measured upon the
microscope, and measure the divisions of the stage-micrometer,
with the spider-line micrometer. Reduce to absolute measure-
174 FOCI AND APERTURE OF OBJECTIVES.
ments from the magnitude of the parts of the micrometers, or if
these are not given, determine them from, the Dividing Engine,
Experiment No. 21. This reduction may be avoided by using two
similar eye-piece micrometers, A and B. Measure several divi-
sions of A with J?, and call the mean of the readings m. Meas-
ure, in the same way, B with A, and call' the mean reading mf.
The true reading, w, will be the mean proportional of these two.
Of course if the micrometers are precisely alike, m will equal mf.
Now measure the distance between them, and call the distance D.
Then if/ equals the focal length of the objective, and/',/" its
111 f
conjugate foci,/' +/" = Z>, jr + jr, = j, and V, = n. From
which /— D, _|_ j>2, and knowing D and ?i,/may be deduced.
The number given by the maker is generally greater than the true
focal length of the objective, and this experiment affords an excel-
lent means of correcting it. To show the value of such measure-
ments, and the accuracy attainable by them, see an article by Prof.
Cross, Journ. Frank. Inst., Vol. LIX., p. 401. Useful work might
also be done by varying Z>, and noting the effect on / also by
changing the correction for cover, or distance between the lenses.
To measure the angular aperture of an objective, screw it into
the end of the tube attached to the graduated circle, set a plate of
glass on the ledge, and bring the objective against it. The front
surface of the lens will then be just over the axis of the circle.
Now clamp the tube, remove the plate of glass, and slide the eye-
piece or small compound microscope into place. Bringing it near
the objective, an image of outside objects is seen, the whole in
fact, forming a telescope with the objective for an object-glass.
The field of view is seen to be bounded by a circle whose true
angular diameter gives the aperture of the objective. Select some
strongly marked vertical line, as the sash of the window, and notice
that as the objective is turned from side to side, the image of this
line moves also. Bring it to coincide first with one edge of the
circle, and then with the other. The difference in the reading of
the index in the two cases equals the angular aperture. Repeat
this measurement with several other objects.
TESTING PLANE SURFACES. 175
83. TESTING PLANE SURFACES.
Apparatus. A stand carrying two telescopes, which may be
placed opposite each other, or set at right angles. The eye-piece
of one, which acts as a collimator, is replaced by a plate of brass
pierced with a very fine hole. This is placed exactly at the focus
of the object-glass, and being illuminated by a lamp, forms a blight
point of light or artificial star. The optical circle might be used
for this experiment, but the graduated circle is not needed, and it
is better to have telescopes of larger size. A millimetre scale is
also wanted, a prism, a sextant-glass, a piece of plate or window
glass, and a lens of very long focus.
Experiment. Make the same adjustment for parallel rays as is
described in Experiment 72. That is, focus the observing telescope
carefully on some distant object as a star, and turn it toward the
collimator. An image of the hole or artificial star at the further
end of the latter will now be visible, but it will generally be out
of focus. Draw it towards, or from its object-glass until accurately
focussed, when it should appear as a very minute circle of light,
like a star. Measure with the millimetre scale the distance be-
tween two points, one on the eye-piece, the other on the end of
the tube in which it slides. Throw the star out of focus by mov-
ing the eye-piece, and focus again ; repeat ten times, and take the
mean of the distances between the two points. Now set the teles-
copes at right angles, and place the surface to be tested at the
intersection of their axes, equally inclined to each, and vertical.
The image of the star reflected in the surface, will then fall in the
centre of the field, and if the surface is perfectly plane will be as
distinct as that previously obtained, although fainter. In gen-
eral, however, it will be a little out of focus, due to the curva-
ture of the surface. In this case move the eye-piece, focus ten
times as before, and take the mean reading of the distance between
the two marked points. Measure the focal length of the observ-
ing telescope, or the distance from its object-glass to the cross-hairs,
also the angle between the axes of the collimator and observing
telescope, unless this is fixed at 90°. Call F the focal length, d
the change in position of the eye-piece, or difference of the means
of the two sets of ten observations, v the angle between the axes,
a the distance from the objective of the telescope to the plane sur-
176 TESTING PLANE SURFACES.
face, and R the required radius of curvature. Then R =
2 • 7— - — ; — , or as d is generally very small compared with
a cos MJ j j
F*
F, it is often sufficiently accurate, if v = 90°, to take R = 2.83— j—
If the surface is concave, the eye-piece has to be pushed in, if con-
vex, out. Test in the same way the other plane surfaces, also the
two sides of the lens. Any distortion of the image is due to
irregularity of the surfaces, as is well shown by trying a piece of
window glass.
The parallelism of two plane surfaces, like those of the sextant-
glass, is well tested in the same way. If both surfaces are per-
fectly plane and parallel, only a single image is formed, otherwise
there are two, one from each face. The angle A, between them,
may be determined from the divergence of the images, by the
formula A — „ — — , in which n is the index of refraction, D
~J( COS ^*
the angle between the images, and r the angle of refraction of
the light in the glass. The latter is known from the equation
sin ^v = n sin r, in which v is the angle between the telescopes.
If v = 90°, A = .267 D, and if v = 0°, A = .33 D. If the sur-
faces are curved, it is also possible to determine the curvature of
both, from the two images, but the problem is then much more
complex.
Another method is to place the telescopes opposite each other,
and cover half their object-glasses with the plate to be tested. If
the two surfaces are plane and parallel, no effect will be produced.
If they are inclined, they form a prism, and cause a second image.
If JO is the angular interval between the two images, and A the
angle between the two faces, A = n ^ or if n = 1.5, A =
2Z>. Comparing this formula with that given above, it is evident
this method possesses only about one-seventh the delicacy of the
other, since for a given value of A, the divergence of the two
images is only a seventh part as great. The delicacy of the
method by reflection may also be increased indefinitely by increas-
ing v. If the surfaces are curved they act like a lens, and throw
the image out of focus. The problem now becomes indeterminate,
TESTING PLANE SURFACES. 177
as there is only one equation to determine the curvature of both
faces. The focus of the equivalent lens may, however, be found
by measuring, as before, the change in position of the eye-piece,
when the focus / will equal - -g — — -. If d is very small,
F*
f = ~jfi which is the best method of measuring the focus of a
very flat lens. Thus, if F = 24 inches, and d = one inch,/ will
equal nearly fifty feet. As in the case of reflection, any irregular-
ity of the surfaces produces a distortion of the image. Test in
this way the plates of glass, and the lens.
Still another method of testing the flatness of a glass plate is to
form Newton's rings, using the monochromatic light of a soda
flame. Very slight irregularities in the surface will then appear
covered with yellow and black rings, like contour lines.
As it is often desirable to increase the reflecting power of a
plane surface of glass when it is to be used as a mirror, the most
common methods of silvering are here appended. A looking-glass
is made by covering the back of the glass with an amalgam of
mercury and tin, as follows. Lay a sheet of tinfoil the size of
the glass to be silvered on a level surface, and pour some mercury
upon it, making it spread over the whole surface with a hare's foot.
Lay a sheet of paper on it, and the glass over all. Then draw the
paper slowly out, when the mercury, as it is exposed, will unite with
the glass, and the paper will remove any adhering dust. Special
care is needed that the tin, mercury, paper and glass, should be
perfectly clean, and that no bubbles remain under the glass. Some-
times the paper is dispensed with, and the glass slid on over the
mercury, bringing it first in contact at one corner. It is then sub-
jected to pressure, and set up on edge to drain. It is best to
keep this mercury by itself, as if used for other purposes, it is
difficult to remove the tin, which gives much trouble by adhering
to any glass surfaces with which it is brought in contact. When
a bright light is viewed in such a mirror, holding it very obliquely,
a large number of images is seen. The first, reflected from the
front surface is faint, the second from the mercury is strongly
marked, and these are succeeded by many others, caused by suc-
cessive reflections, and growing fainter and fainter until they fi-
nally become invisible.
12
178 TESTING TELESCOPES.
To obtain a single image only, sometimes a plate of black glass
is used, or the lower surface is covered with black paint, or better,
since much light is lost in this way, the front surface is covered
with a deposit of metallic silver. One method of doing this is by
dissolving 10 grms. of pure nitrate of silver in 20 grins, of water,
and adding 5 grms. of ammonia. Filter, add 35 grms. of alcohol of
specific gravity .842, and 10 drops of oil of cassia. Cover the plate
with this mixture to a depth of quarter or half an inch, and add 6
to 12 drops at a time of a mixture of 1 part of oil of cloves to 3 of
alcohol, until the whole surface is covered with the precipitated
silver. The plate is then dried, cleaned and polished. Various
other receipts are recommended, some using starch, sugar, or tar-
taric acid, instead of oil of cloves to precipitate the silver. Proba-
bly much more depends on the practice and skill of the experi-
menter than on the details of the different formulas. Liebig em-
ploys a liquid formed by adding soda-ley of sp. gr. 1.035 to 45 cm.8
of an ammoniacal solution of fused nitrate of silver, and dissolving
the precipitate formed by adding ammonia until the volume equals
145 cm.8. Add 5 cm.8 of water, and shortly before using it, mix
with one sixth to one eighth of its volume of a solution of sugar of
milk, containing 1 part of sugar, to 10 of water. Flood the glass
to a depth of half an inch, and it will soon become covered with
a thick coating of silver.
Another method of making reflectors, is by platinizing glass, or
covering it with a layer of metallic platinum. This is accom-
plished by covering the surface with chloride of platinum with a
brush, reducing it to a metallic state by oil of lavender, and heat-
ing it in a muffle.
84. TESTING TELESCOPES.
Apparatus. A long darkened chamber with a small aperture
at the farther end, through which the light of the sky, or of a lamp,
shines. A long empty water-pipe, or unused flue, answers very
well for this purpose, but if this cannot be obtained, a large black
screen with a small hole in it may be placed at the farther end of
the room, and a short tube blackened on the interior, used to cut
off the stray light. A double length may be obtained by placing
a plane mirror at the farther end of the room, and the screen close
to the observer. A telescope to be tested, which should have an
TESTING TELESCOPES. 179
object-glass at least three inches in diameter, is also needed. It is
composed of two lenses, one concave, of flint, the other convex, of
crown glass. The focus of the latter will be about three-fifths
that of the two together. Suppose this is three feet, then the
focal length of the crown glass will be about 22 inches. Procure
two similar lenses of 20 and 24 inches focus, respectively. Com-
bining the first with the flint gives an under-corrected, while the
other gives an over-corrected lens.
Experiment. The principal defects to be sought for, are striae
or irregularity of the glass, spherical aberration or incorrect form,
chromatic aberration or imperfect correction for color, imperfect
annealing of the lenses, and wrong centering or want of coinci-
dence of their axes with that of the telescope.
First, to test for striae, direct the telescope towards the artificial
star or minute point of light at the farther end of the room. Then
remove the eye-piece, and placing the eye in the axis of the in-
strument a bright circle of light will be seen, which will cover the
whole object-glass when the eye is exactly at the principal focus.
If now the glass is free from veins, striae, or other imperfections,
this circle will appear perfectly uniform, otherwise the striae are
shown in a very marked manner. To determine whether they are
caused by the crown or flint lens, remove the latter, and see if
they still remain. Test in the same way the other two convex
lenses, and sketch any striae that may be present. Some cheap
cosmorama lenses are made of common plate-glass, in which case
they are often full of parallel striae, running in the direction in
which the glass was rolled.
To test for spherical aberration, place the eye a little beyond
the focus, and pass a card through the latter. Since all the rays
would intersect at the focus if there were no spherical aberration,
the light would be instantly extinguished when the card covered
this point. In practice, however, the bright circle of light assumes
the appearance of a curiously shaped surface of revolution, from
which the form of the lens is readily determined.
To test for chromatic aberration, examine the image of the
artificial star with an eye-piece, precisely as when looking through
the telescope at a real star. If the lens was perfectly achromatic,
a very minute circle of light would be obtained, which would
enlarge on pushing the eye-piece in or out, remaining all the time
180 TESTING TELESCOPES.
perfectly colorless. The change in size is due to the fact, that the
rays of light form a cone of which the object-glass is the base, and
the focus the apex. The field of view is really a section of this
cone at right angles to its axis. If an uncorrected lens is used,
since the violet rays are more bent than the red, they form a cone
with vertex nearer the object-glass; accordingly, if the eye-piece is
pushed in, the centre of the circle will be violet, and the exterior
red. Owing to the unequal dispersion of different parts of the spec-
trum by the two glasses, it is impossible ever to obtain entire free-
dom from color, but the best correction is obtained when the eye-
piece being pushed in, the circle has a bluish purple exterior, and
when pulled out, a lemon green exterior. In the same way an un-
der corrected lens should give inside the focus a pure purple, and
outside a yellowish margin ; an over-corrected lens will give a blue
or violet color inside, and outside an orange margin. Use the three
convex lenses in turn, and note the colors in each instance. Many
other things may be learnt from the appearance of the artificial
star. Thus if part of the object-glass is covered, the circle assumes
the shape of the uncovered portion. Spherical aberration also
shows itself by the formation of rings in the image of unequal
brilliancy, and imperfect centering or obliquity of the lenses, by
converting the circle into an ellipse, or throwing out a ray of light
on one side. This effect is greatly increased by using the mono-
chromatic light of a soda flame.
One other test remains to be applied, that for imperfect anneal-
ing of the glass. Lay a plate of glass horizontally in front of the
window, so that the light reflected from it shall be polarized. In-
terpose the lens between it and the eye, and examine the transmit-
ted light by a Nicol's prism, as will be described more in detail in
Experiment 88. Any inequality in density of the glass will at
once show itself by dark patches, which change their position as
the prism is turned. Of course all these tests must be re-
garded as secondary to the real test of every large telescope by
trying it on various celestial objects of known difficulty, and com-
paring the results with those obtained with other instruments of
the same size.
Next measure the magnifying power of the different eye-pieces
furnished with the telescope. The power of a small telescope or
PHOTOGRAPHY. 181
opera-glass is readily measured by looking simultaneously at the
object with one eye, and at its image with the other, and compar-
ing their relative magnitudes. The best object to be used is a
large scale of inches, noting how many divisions as viewed by
the eye, equal one or two as seen through the telescope. For high
powers the best method is by the dynameter. Focus the telescope
on a distant, object, and turn it towards the sky, or other bright
light. On holding a sheet of paper near the eye-piece, a bright
circle of light is seen projected on it, which is really an image of
the object-glass formed by the eye-piece. The diameter of the
object-glass, divided by that of its image, equals the magnifying
power. To measure accurately the diameter of the small circle, a
spider line, or eye-piece micrometer, may be used, or a small read-
ing microscope, whose objective is divided in two parts, which may
be moved past each other a known amount by a micrometer-screw.
Two images of the circle are thus formed, which may be rendered
tangent to each other by turning the screw. The parts of the
objective have then been moved a distance proportional to the
diameter of the circle, which is thus measured with great preci-
sion. In cheap telescopes a diaphragm is sometimes inserted near
the objective, thus reducing the available aperture, and increasing
the sharpness of definition of a poor lens, though diminishing the
amount of light without apparently reducing its size. This is
detected by turning the eye-piece towards the light, and seeing if
it is visible when looking through the very edge of the objective.
If not, the diaphragm should be removed or an incorrect value of
the magnifying power will be obtained. If focussed for near
objects, the magnifying power is much increased; hence for pur-
poses of comparison, objects at a great distance should always be
selected when making this measurement.
85. PHOTOGRAPHY. I. GLASS NEGATIVES.
Apparatus. A small darkened chamber or closet is needed for
this Experiment. In this, a sink is placed with an abundant
supply of water, and over it a shelf for the bottles containing the
various re.'igents described below. A glass or porcelain vessel,
shaped somewhat like a card-case, is employed to hold the solution
of nitrate of silver, and a flat dish for the hyposulphite of soda.
A large number of plates of glass are needed on which the
182 PHOTOGRAPHY.
photographs are to be taken, and racks for holding them. In
preparing the solutions, funnels, filter-paper, and other similar
chemical apparatus are also required. The closet should be
lighted by a gas-burner covered with a yellow shade, to cut off
the actinic rays. The camera in which the photographs are taken
consists of a blackened box with a convex lens in front, and closed
behind by a plate of ground glass, or by the plate-holder which
carries the prepared plate of glass.
Great care has been bestowed on various forms of lenses for
cameras, and the best forms are somewhat expensive. Three
kinds are commonly employed. First, portrait lenses which have
a large aperture, and admitting much light work very quickly, but
they only take in a cone with an angle of about 60°, and have not
much depth of focus. That is, when focussed for a given distance,
objects a little nearer or a little farther off will be indistinct. The
second kind of lens is adapted for views ; a small concave lens is
inserted between two which are convex, thus giving a greater
depth of focus, but not working so quickly as the preceding. The
third class, as the globe and the Zentmayer lenses, take in a large
cone of light, as 90°, but work very slowly, requiring one or two
minutes even in strong sunlight. They have a great depth of
focus, andmay be placed very near the object; but when this is a
building, the perspective will be bad if the camera is brought too
near. For the same reason, when taking a building, the glass
plate should always be vertical, or a distortion will be produced.
As no lens is perfectly achromatic, the focus of the actinic rays
will not coincide exactly with the visual focus, being less for an
under, arid more for an over-corrected lens. The latter should
always be avoided, and it is best to get one in which the two
foci coincide as nearly as possible.
Experiment. Almost all photographic processes depend on
first taking a glass negative, that is, a picture in which the bright
portions shall be transparent, the dark parts, opaque. For this
purpose the plate of glass is first prepared or rendered . sensitive,
then exposed in a camera obscura so that an image of the object
shall fall on it. The plate is then developed, by which the image
is rendered visible, and fixed or rendered permanent. These
operations are easily performed when all the apparatus is ready,
and the baths employed in good condition. The real difficulty in
taking photographs, is in preparing the various solutions used, and
in renewing them as they deteriorate ; accordingly the following
receipts are given, which the student should, if possible, try for
himself.
PHOTOGRAPHY. 183
The collodion used for coating the plates is made as follows.
To 8 oz. of ether add 96 grains of gun-cotton, and then 8 oz. of
alcohol. The strength of the latter must be 95 per cent., and it
must contain no fusel oil, or the cotton will not be well dissolved.
If made in the evening be very careful about lights, as the mixture
may take fire from a lamp several feet distant. As the vapor is
much heavier than air, there is more danger from lights on the
floor than from those above. Dissolve 24 grains of bromide of
potassium, in as little water as possible, add 64 grains of iodide of
ammonium, and more water if necessary, and pour this into the
collodion to iodize it. Pure iodide of ammonium will do, but it is
better to have the iodine a little in excess, in which case, the salt
will be dark colored instead of white. Too much iodide and too
little bromide give a hard picture. In about two days the
collodion will be ready for use, and it will keep in good condition
about two weeks. Iodide of cadmium, which is commonly used
in that which is sold, makes collodion keep better, but renders it
less sensitive. If the weather is very hot, there is difficulty from
the collodion drying too rapidly. In this case, less ether and more
alcohol must be used, as the latter is much less volatile.
The silver-bath by which the plate is rendered sensitive, is
made by adding 1 oz. of nitrate of silver to 12 oz. of water, and
acidulating it with 30 or 40 drops of pure nitric acid. If used
directly it would dissolve the iodides in the plate. ^Accordingly,
a coated plate should be left standing in it over night. Filter,
and in twenty-four hours it can be used. More depends on the
condition of the silver-bath than on that of any other liquid
employed. It should be kept nearly neutral, but always slightly
acid. If alkaline the picture is fogged or blurred, and if too acid
the action is much retarded. Organic matter is very injurious,
and dust should therefore be carefully excluded by a cover.
Whenever practicable, it should be exposed in a glass bottle to
strong sunlight, which precipitates the organic matter in black
flakes, and the latter removed by filtering. After using the bath
for some time, it becomes covered with a scum, due to the alcohol
and ether from the collodion plate, which is the first indication
that the solution is becoming too weak. The iodide of silver is
then liable to be precipitated on the plate, forming little spots in
184 PHOTOGRAPHY.
it like pinholes. One half its bulk of water should, in this case,
be added, to precipitate the iodide, the solution filtered and boiled
down to its original strength.
To develope the picture when taken from the camera, a solution
of 1 oz. of proto-sulphate of iron, in about 20 oz. of water is used,
and containing 1 or 2 oz. of pure acetic acid, No. 8. This liquid
will keep only about three days. The object of the acetic acid is
to retard the 'process, as otherwise the silver would blacken
instantly.
The liquid used to fix the picture, is formed by dissolving 1 oz.
of hyposulphite of soda in 5 oz. of water.
Be careful that the glass is not rusty or iridescent, as in that case
the collodion is liable to cleave off. Double thick glass is prefer-
able on account of its greater strength, unless the plates are
small. There must be no " knobs " or glass dust on its surface, nor
deep scratches, because these will appear in the positive if the
printing is done by sunlight. Almost all plates are slightly
curved, and it is essential that the hollow or concave side should be
coated. The plate will then conform more nearly to the image in
the camera, is easier to coat, is less liable to be scratched if laid
down on its face, and is not so likely to be broken in printing.
To clean the glass, mix equal parts of alcohol and ammonia, and
add enough rotton-stone to render it viscid. Pour a few drops on
the glass and scour with wash leather, letting the plate rest on one
corner. Clean also the edges carefully. Let it dry, and then rub
off all the rotton-stone with clean flannel. Breathe on the plate,
and if clean, the moisture will pass off rapidly and evenly from
the surface. Thus cleaned, they will keep for two days. A
better method is the following. Soak the plates over night in
strong nitric acid, and wash thoroughly under a faucet. Mix up
the white of an egg with an egg-beater, taking care that none of
the yolk gets in, add 12 ounces of water and 3 or 4 drops of
concentrated ammonia, which keeps the egg from souring, and
neutralizes any acid that may remain on the plate. Filter through
paper or cloth and keep the filtrate in a bottle. Pour it over the
plate, and a uniform film is produced, which will last for six
months if kept dry and free from dust. Another method is to
soak the glass over night in a strong solution of caustic potash
PHOTOGRAPHY. 185
and then wash it, but this is liable to injure the silver bath. If
any nitric acid gets in, it merely retards the action, but alkali will
cause fogging, or the image will look smoky, as if under a veil.
To coat a plate properly with collodion requires considerable
skill and practice, as it is very essential that the coating should be
perfectly uniform. First, remove any films of collodion that may
have dried on the mouth of the bottle, and take care never to
disturb the sediment at the bottom. Regarding the plate as a
map, it is held by the &. W. corner with the thumb and forefinger
of the left hand, and the collodion poured on just N". E. of the
centre. Enough is added to cover about half the plate which is
then inclined, so that the liquid shall flow successively to the
N. E., the N~. TK, the &. Wl and the /& E. corners, and then tipped
so that it will run off into the bottle. Allow it to drain for a few
seconds and incline it gently from side to side, to prevent its
forming streaks. The ether soon evaporates, and as soon as the
film becomes sticky and consistent, the plate is immersed in the
silver bath, by laying it on its holder, and lowering it into the
liquid. This must be done slowly and steadily, or streaks will
appear across the plate. If the plate is too large to be held by
one corner while coating it, lay it on the palm of the hand, and
interpose a sheet of card-board to prevent the warmth from drying
the collodion too rapidly. Still larger plates must be placed on a
board and rested on a point attached to the top of a tripod.
After remaining a few seconds in the bath, the plate should be
raised gently out of it, when its surface will present a greasy
appearance, due to the ether still remaining in the film ; soon
however, it will appear to be perfectly wet by the solution, and is
then ready to be transferred to the camera. The plate-holder
consists of a frame closed in front by a slide, and with a hinged
back. The latter is opened, the plate put in with the coated side
next the slide, and the latter then closed. A black cloth is thrown
over it to cut off any stray light that may leak in through cracks
in the holder, and it is then carried to the camera. The object to
be taken having been placed in the proper position, and in a good
light, the camera is turned towards it, when an image will be
formed on the ground glass. This is best seen by standing behind
the instrument, and cutting off the light by a black cloth thrown
186 PHOTOGRAPHY.
over the head. The size of the image may be varied by altering
the distance of the camera, and the latter focussed by changing
the distance between the ground glass and lens. For near objects
the focus is greater and the size of the image larger than for
those more distant. The image may be brought to the centre
of the glass, by turning the camera or inclining it. When the
image is satisfactory and carefully focussed, remove the ground
glass, and replace it by the holder. Cover the lens with a cap or
black cloth. Then draw the slide of the holder, and when all is
ready, expose the plate to the light by removing the cap. The
time of exposure varies with the light, the object, arid the kind
of lens, and must be learned by experience. When the time has
expired, replace first the cap, and then the slide, and taking out
the holder carry it into the dark room. Take care and never turn
the plate-holder over after the plate is in it, as the silver collects
along the low.er edge and would, if inverted, flow over the glass,
forming streaks.
On taking the plate out of the holder no trace of the picture is
visible, but the film merely appears of a creamy white throughout.
Pour on some of the developer so as to cover the plate at one
flow (or streaks will be formed), holding the glass horizontal with
the prepared side up, and in a few seconds the picture will appear,
the portions acted on by light turning dark, the others remaining
unchanged. The plate is then washed under the faucet to remove
the developer. If the exposure has been too long, the picture will
develop instantly, giving a dense blurred negative. With too
short an exposure or too feeble a light, a faint transparent picture
is obtained, which developes only very slowly. In this case, it
may be improved by re-developing, or pouring on a mixture of
the silver bath and developer as when developing. Pryrogallic
acid is sometimes used instead of sulphate of iron.
The picture is then rendered permanent by immersing it in the
solution of hyposulphite of soda, which dissolves the iodide of
of silver, where unacted on by light, rendering these parts of the
plate transparent.
It is still easily scratched, and should be varnished if it is to be
handled much. For this purpose, it is warmed over a lamp until
hot to the touch, and amber varnish poured on precisely as when
PHOTOGRAPHY. 187
coating a plate. If not warmed, the varnish gives a precipitate.
Be careful that it does not catch fire, as it then dries in ridges.
Warm again gently, to harden the varnish. The completed
negatives are best kept side by side and vertical, in racks.
The most common difficulty in taking photographs is fogging,
or the picture appearing misty or indistinct. In this case, test the
silver bath, and if it is alkaline add a little nitric acid. If too
much is added the process is retarded, and it must be nearly
neutralized with ammonia. Great care must be taken that the
plate is not exposed to stray light before or after placing it in the
camera, and that the latter and the plate-holder have no cracks in
them, through which the light may enter.
When taking views, the camera should never face the sun if it
can possibly be avoided. If this is unavoidable, hold a hat or
board so that its shadow shall fall on the camera, thus cutting off
the direct rays of the sun from the lens. When light and dark
objects, as the sky and deep shadows, are to be taken at the
same time, hold a screen in front of the lens, to cut off the brighter
objects until near the end of the exposure, and then remove it.
As objects for the stereoscope or lantern, glass positives are
needed, that is, photographs on glass, in which the light portions
shall be transparent, and the dark parts opaque. A negative is
first taken and then photographed like any other object, reflecting
the light of the sky through it by a sheet of paper, and cutting off
all stray light in front. Porcelain photographs were formerly
taken in this way, but both are now often printed like paper posi-
tives. Ambrotypes are negatives exposed for a short time, so that
the dark portions are very transparent, and are rendered black by
placing a sheet of tin covered with black varnish behind them.
86. PHOTOGRAPHY. II. PAPEB POSITIVES.
Apparatus. Some albumenized paper, and the reagents needed
to render it sensitive and to make the picture permanent. Also
some negatives to be copied, and several printing-frames in which
to expose the paper to the light. Strong sunlight is requisite for
this Experiment, and it is much better to place the printing-frames
outside the window, rather than let the sunlight first pass through
the glass. Three flat porcelain dishes are needed to hold the
188 PHOTOGRAPHY.
liquids in which the paper is immersed, spring clothes-pins by
which to hold it when drying, and an abundant supply of water
in which to wash it.
Experiment. It is difficult to judge of the true appearance of
an object from the glass negative, as the lights and shades are
inverted, and, moreover, several copies of a photograph are often
wanted for distribution. It is therefore customary to print
positives on paper, by the process described below. A fine grained
paper is employed, covered with a thin layer of albumen, which
fills up the pores and gives a good surface. Excellent albumen-
ized paper may now be procured ready-made, as it is manufac-
tured on a large scale, for photographic purposes. When the
picture is to be touched up with India ink, or colored, common
paper is often preferred, as the paint will not adhere to the
albumen. To render it sensitive, a solution of nitrate of silver is
prepared, of a strength of about 40 grains to the ounce, but
depending on the amount of chloride of sodium in the paper. It
should also be somewhat stronger in summer than in winter.
Render it alkaline by a few drops of ammonia, and pour it into a
flat porcelain dish. If any scum appears on the surface, remove it
by a piece of tissue paper.
Take a piece of the paper, somewhat smaller than the dish, by
the two opposite edges, with the albumenized side down, and dip
it into the dish so that the centre line shall touch the liquid.
Lower the edges so that the paper shall float on the liquid, taking
care that no air-bubbles are imprisoned under it, or they will form
white circles on the picture. Breathe on the edges of the paper
to prevent their curling up, by the warping due to the expansion
of the lower side, and take great care that no silver gets on the
upper side of the sheet. The time of immersion must be found
by trial ; it is generally one or two minutes. If too long, the sil-
ver works through the albumen into the paper, turning it yellow
on exposure, while if too short, the salt does not decompose and
fill up the surface with the silver, so that a mottled appearance is
produced. To obtain the best results, various devices are em-
ployed, such as fuming the paper with ammonia, which gives a
clearer picture. After taking the paper out of the bath it should
PHOTOGRAPHY. 189
be hung up by the corners by spring clothes-pins, in the dark. It
will only keep a short time, in winter for a day or two, in summer
scarcely over night. The prepared surface should never be
touched by the fingers, or their mark will appear in the finished
picture.
The paper must now be exposed to sunlight under the negative.
The pressure frame in which this is effected is made somewhat like
the frame of a picture, only it is much more substantial, and so
arranged that the back can be removed; the latter, when in
place, is pressed strongly against the glass in front, by springs.
The back is lined with felt, and hinged, so that one half can be
turned back, and part of the picture examined without disturbing
the remainder. Having cut the paper of the right size, the pres-
sure-frame is placed face downwards, and the back removed. A
plate of glass is first inserted, like the glass of a picture, then the
negative with the prepared side up, next the paper with the al-
bumenized side down, and finally the back of the frame is restored
to its place. Great care must be taken always to place the pre-
pared sides of the glass and paper in contact, as otherwise only a
blurred picture will be obtained.
Now turn the frame over, and place it in strong sunlight, and
inclined at such an angle that the light shall fall on it nearly nor-
mally. It will soon be seen that the unprotected portions of the
paper, that is, those under the transparent parts oi the negative,
are turning under the influence of sunlight from white to black.
After a few minutes, take the frame out of the light, and opening
half the back, bend the paper so as to examine it, when the pic-
ture will appear in its true aspect with shades dark, and the bright
portions, light. If the picture is not dark enough, replace the back,
and thus proceed, until the paper is somewhat darker than is
desired for the finished photograph. The picture thus obtained is
of a purplish color, and if exposed to light would turn completely
dark. It must therefore first be toned, or its color altered, and
then fixed, or rendered permanent.
The bath for toning is made of a solution of chloride of gold.
Procure some pure gold from a gold beater, dissolve it in aqua
regia and evaporate to dryness. Dissolve 5 grains of the yellow
chloride of gold thus obtained, in 1 oz. of water, and neutralize it
190 PHOTOGRAPHY.
with carbonate of soda. This must then be mixed with about ten
times its bulk of water, using a stronger solution in cold than in
warm weather. Wash the picture thoroughly in running water
for five or ten minutes, to remove all the free silver which would
otherwise precipitate the gold and give a foggy picture, and then
dip it face downwards into the toning solution. The effect of this
will be to alter its color until a certain reddish blue tint is attained.
The best effects are obtained with a lukewarm solution, of such a
strength as to tone in about ten minutes. If not toned sufficiently
the color is reddish, and if this process is carried too far it takes
the life out of a picture, and gives it a chalky appearance. In the
same way, too rapid a toning gives a mealy look to the photo-
graph. As the paper withdraws the gold gradually from the solu-
tion, rendering it weaker, more should be added, first neutralizing
it with soda. It is well to move the bath gently, swaying it from
side to side. Be very careful that no hyposulphite of soda gets
into the gold solution, as it would spoil it, forming a precipitate.
As the paper is still sensitive to light, the toning should be carried
on in a dark room, just light enough to distinguish clearly the
color attained. To stop the toning, the photograph is next washed
in cold water to remove the gold ; it is then rendered permanent
by immersion in a solution of hyposulphite of soda, 1 oz. of the
salt to 8 oz. of water. As the picture is rendered lighter and red-
der by this process, allowance must be made, in the printing and
toning. The photograph must now be washed very thoroughly in
running water for an hour or two, or better, over night, to remove
every trace of hyposulphite. Otherwise a black sulphide of silver
is formed, which turns in time to red, and spoils the picture. To
see if all the hyposulphite is removed, hold the paper up to the
light, when it should appear clear ; if mottled, the washing must
be continued.
It now only remains to mount the photograph, that is, to paste it
on to cardboard, to make it flat and stiff. Lay it face upwards on
a plate of glass, and lay on it a second plate, of the size to which
it is to be cut. It is thus easily centered, and cut to the right size
by drawing a sharp knife along the edges of the glass, holding the
blade always at the same angle of inclination. Attach it to the
cardboard by paste made of the best wheat starch. The latter is
TESTING THE EYE. 191
first moistened with cold water, and boiling water then poured
upon it ; if it does not come thick enough it may be heated for a
moment to the boiling point, but should not be boiled, otherwise
it becomes watery and loses its adhesiveness. To prevent the
cardboard from curling up, it is well to moisten it before attaching
the photograph. The whole should then be passed through heavy
rollers to give a good finish to the picture.
87. TESTING THE EYE.
Apparatus. A set of test-types, or letters of various sizes, should
be placed at distances from the student proportional to their size.
On the table are placed the optometers described below, a reading
microscope on a stand, for the experiment of Cramer and Helm-
holtz, a gas flame, tests for astigmatism, and a set of concave,
convex and cylindrical lenses of various curvatures, and prisms of
various angles.
Experiment. The eye is formed like a camera obscura, in
which the retina takes the place of the screen on which the image
is received. In front of the lens is a delicate curtain, called the
iris, which gives to the eye its color, and in this is a circular hole,
the pupil. The iris is formed of fibres, some circular, others
radial, the contraction of the first diminishing, of the second
increasing the size of the pupil, and hence the amount of light
admitted into the eye. These changes are readily seen by cover-
ing the eye with the hand, removing the latter, and looking in a
mirror; the pupil will then be seen to contract slowly, having
dilated in the dark. The pupil of the other eye will also contract
a little, as they both commonly act together.
The image of objects at various distances, may be brought to a
focus on the retina by varying the form of the lens, while in the
camera it is effected by varying its position. By this change
which is called accommodation, objects may be seen with perfect
distinctness, with a normal or perfect eye, at any distance, from
about 4 inches to infinity. Call P the nearer distance, and R the
farther, for any eye, then ~7 — "p — ~R '1S ca^e<^ tne range of
accommodation, and is much employed in studying defects of the
eye. In the case of the normal eye, the range of accommodation
192 TESTING THE EYE.
evidently equals £. The most common defect to which the eye is
subject, is that the ball is not spherical. If it is elongated, the
retina is carried too far off, and objects must be brought nearer
the eye, to render them distinctly visible. Such an eye is called
myopic, or near-sighted. If the ball is flattened, near objects
cannot be easily seen, and the eye is then hypermetropic, or far-
sighted. This must not be confounded with the effect of age,
which renders the lens harder and thus diminishes the range of
accommodation, so that distant objects alone can be seen. The
eye is then said to be presbyopic. The normal eye is called
emmetropic.
To measure the far and near points optometers are used. One
of the simplest of these consists of a board on which a straight
line is ruled. At one end is a sheet of metal, with two fine slits
very near together. The eye is placed close to the slits, so as to
look through both, when it will be noticed, that the nearer end of
the line appears double, since the images formed by the two slits
cannot be brought together by the eye, on account of the short
distance. Sometimes, also, the farther end will appear double, if
the eye is myopic. The points, where the line divides, give the
far and near limits of distinct vision. A better form of opto-
meter, resembles the apparatus represented in Fig. 60, only it is
much smaller. The lens, which should have a focus of 6 inches, is
fixed at the end of the rod, in the place of the gas-burner, A, and
some very fine print, or other minute object, is attached to the
screen, C. Now measure the greatest and least distance at which
the print can be read, when the eye is placed near the lens. Call
these distances Pf and JRf. Then -p = -p- — -g-, and j? = ^
— -g-, which gives P and _K, the far and near distances of accom-
modation. For the normal eye, as P = 10, R — oo , P' should
equal 2.67, and E = 6.
Another excellent form of optometer is very simply made of a
sheet of cardboard or brass. This is pierced with three sets of
holes, the first a single hole 1 mm. in diameter, the second several
smaller holes near together, for instance three rows of three each
at intervals of a millimetre, and thirdly two holes 3 mm. apart,
TESTING THE EYE. 193
one of which may be covered with a plate of red glass. View a
small distant point of light, as a candle or gas flame, through
them, and the appearance will vary according as the eye is normal,
far or near sighted. Looking through the single hole and moving
the card rapidly from side to side, the light will appear to remain
stationary, if the eye is normal, otherwise it will appear to move
as the rays pass through different portions of the pupil. In the
same way the nine holes will give nine images, if the eye is not
normal. If the two holes are used, two circles are seen which
overlap as the card is brought near the eye. If the eye is not
normal two images will be formed in the overlapping part, since
the rays falling on different parts of the lens are not brought
together to the same point on the retina. Now cover the right
hand hole with the red glass, and if the eye is far-sighted the left
hand one will be colored, if near-sighted the right one, since in
the latter case the rays cross, coming to a focus before reaching
the retina. From, the distance between the images, the amount
of the defect may be measured. Thus bring a second candle
flame near the first, until two of the images overlap, forming three
instead of four ; the distance between the candles then equals the
interval between the images, and from it the lens required to
render the eye normal, may be determined. Let D be the interval
between the images, d that between the two holes, and B the
distance from the light to the eye. Then, D : d =-• JB : ' f in which
f is the focal length of the lens required to produce distinct
vision. By turning the card the twro images will appear to re-
volve around each other, and if the eye is astigmatic their distance
apart will vary. If the eye is normal all- these effects may still
be observed by putting on convex, concave, or cylindrical glasses.
When observing one's own eye it is often more convenient to
view the reflection of two lights near at hand in a distant mirror,
so that their distance apart may be more easily varied.
The best test, however, for the eye, is to see if all the test-letters
can be read easily. To understand how objects look to a near-
sighted person, put on a pair of convex glasses, and repeat these
observations with them. Do the same with concave glasses, which
give the effect of hypermetropia. See also if the foci of the glasses
can be determined correctly from these observations.
13
194 TESTING THE EYE.
Another defect present in some eyes is astigmatism, or unequal
focus for horizontal and vertical lines. For example, the eye may
be normal for vertical, and near-sighted for horizontal lines. It
is detected by looking at a test made of several series of strongly
marked equidistant lines, running in various directions. This
defect is corrected by using cylindrical lenses, or if, as often hap-
pens, the eye is myopic or hypermetropic at the same time, by
means of lenses cylindrical on one side, and convex or concave on
the other. Many persons who could never see well with common
glasses, experience wonderful relief from such lenses. Sometimes
the axes of the eyes are not quite parallel, a defect remedied by
the aid of prisms, with very acute angles.
Many theories have been advanced to explain accommodation,
some supposing that the retina was drawn back, others, that the
lens moved, and others, that the ball of the.
eye changed its shape. The true explana-
tion is deduced from the following experi-
ment, devised and worked out quantitively,
by Kramer and Helmholtz. Two persons
m 62 are required; one, whose eye is to be exam-
ined, sits facing a candle, or gas-burner,
while the other examines with the reading microscope the reflec-
tion of the light in his eye. Three images will be seen, as shown
in Fig. 62, in which V ig intended to represent the reflection of
the candle flame. The eye being directed towards a distant ob-
ject, the first image to the right is formed
by reflection in the cornea, or front surface
of the eye. It is bright and upright, as the
V
v
V
Fig. 63.
\ / surface is convex. The second is formed
by the front surface of the lens. It is much
fainter and larger, but also upright. The
third being formed in the posterior and
concave surface of the lens, is minute and inverted. Now let
the eye be directed towards a near object. The first and third
images will remain unchanged both in size and position, showing
that the cornea and rear surface of the lens are not altered, either
in position or curvature. But the second image, as shown in Fig.
TESTING THE EYE. 195
63, approaches the first, and diminishes in size, showing that the
front surface of the lens is pushed forward, and becomes more
curved. Measurements also show, that the amount of the change
is just sufficient to account for the required difference in focus.
This experiment is very conclusive, as each of the other hypoth-
eses is disproved by it. If the cornea altered, the first image
only should move. If the lens moved, the second and third
images should approach the first without altering their size, and
if the form of the ball altered, the relative position of all three
should remain unchanged.
All parts of the retina are not equally sensitive ; although the eye
can perceive objects through an angle of about 150° horizontally,
and 120° vertically, yet the portion where vision is most distinct
is quite small, not more than 3° or 4° in diameter. This
portion of the retina, which is called the macula lutea, is used
almost exclusively whenever objects are carefully examined, and
probably on this account, is not quite as sensitive to very faint
objects as the adjacent parts. At any rate, it is very customary
with astronomers, when trying to see very faint objects, to direct
the eye a short distance from their supposed place, and try to
catch sight of them, when not in the centre of the field of view.
A short distance from the macula lutea, on the side towards the
nose, is a small circle where the optic nerve enters. This space,
although so near the most sensitive portion of the retina, is
totally insensible to light. It is called the papilla, or some-
times the punctum ccecum, or blind spot. To observe it, mark
two points on a sheet of paper, about 4 inches apart, and closing
the left eye, direct the other to the left hand point, and then
moving the paper to and fro, a certain distance will be found, at
which the other point will completely disappear. By using two
lights, this experiment maybe rendered still more striking, as even
a bright light may be made to completely disappear, although ob-
jects all around it are visible.
A great variety of experiments may be made, depending on the
stereoscopic effects obtained with two eyes, or on the persistence
of vision, using such instruments as the thaumatrope, chromatrope,
and phenakistascope.
196 OPTHALMOSCOPE.
88. OPTHALMOSCOPE.
Apparatus. The instrument known as the " Opthalmoscopic Eye
of Dr. Perrin," is admirably adapted as a substitute for a human
eye, on which to use the opthalmoscope. It consists of a brass
ball, on a stand, representing the globe of the eye, a series of cups,
painted to represent different diseases of the retina, which may be
inserted in its rear portion, and lenses, representing the cornea and
lens, which may be screwed on in front. To these, diaphragms,
representing the change in diameter of the pupil, or aperture in
the iris, may be attached. An Argand burner is needed, and a
Grafe's opthalmoscope, also some plates of glass, and a small mir-
ror, with the silvering removed from a circle a quarter of an inch
in diameter. This Experiment should be performed in a darkened
room, but instead, the light may be cut off by a large screen
of black cloth.
Experiment. The opthalmoscope which is used in studying the
interior of the eye, has caused a complete revolution in this branch
of medical science. Its inventor, Helmholtz, reflected light into
the eye, by a piece of plate glass, and then looking through it,
found the interior sufficiently illuminated to be visible.
Take the model of the eye from its box and place it on its
stand. Insert the retina marked 1, which represents the normal
or healthy retina. Screw on in front the lens marked E. J/., and
light the burner, placing it by the side of the model eye, and
about a foot distant. By reflecting the light into the eye by
a plate of glass, and looking through the latter, Helmholtz'
experiment may be repeated, and a view of the interior obtained.
This is more easily accomplished by using several plates, or better
still, with the mirror from which the silvering has been partially
removed.
The opthalmoscope consists of a circular, concave mirror, with
the silvering removed from the centre. Just behind it is placed
a fork, in which either of the five small lenses may be placed.
They should be numbered on their edges from 1 to 5, the
former being the most concave, the latter the most convex. The
retina of the normal eye is placed at a distance from the lens,
equal to its principal focus, hence its image is formed at an
infinite distance, or the rays emerge parallel. It can therefore be
viewed without any lens, using the mirror precisely as in the
OPTHALMOSCOPE. 197
previous experiment. The image is better seen, if one of the
lenses 1, 2, or 3 is inserted in the fork, as the distance of the image
is then diminished, so that the rays diverge, instead of emerging
parallel.
This method has the objection of showing only a small portion
of the retina at a time, and of bringing the observer too near the
eye for convenience. Another method is therefore more com-
monly used, in which an aerial image is formed in front of the
eye, by a convex lens, and this is viewed either directly with the
eye, or with a second convex lens of long focus.
Hold the large lens, or objective, two or three inches in front of
the model, with the left hand, steadying it with the little finger,
which, in the case of the real eye, rests on the forehead of the
patient. The mirror should be held a foot or more distant, and
turned into such a position as to reflect the light of the gas flame
into the model. After a few trials a very beautiful view of the
retina will be obtained. The image will be inverted, and may be
made as large as the objective by' removing the latter to a distance
equal to its focal length. To view the other portions of the retina,
the model must be turned from side to side, or the patient re-
quested to direct his eye towards various points in turn. The
image is improved by placing lens 4 or 5 behind the mirror. In all
these experiments, if near-sighted, use a lens with a number lower
than that here recommended; thus, instead of 2, use 1, for 5, use
4, etc. The first of the above methods, that is, without the ob-
jective, is called the direct, the second the indirect method.
Now screw the larger diaphragm over the lens, and try once
more to view the image ; then replace it by the small diaphragm,
with which it is about as diificult to observe the retina as with the
eye in its normal condition. The larger diaphragm corresponds to
the case where the pupil is expanded by belladonna. With the
small diaphragm it is easier to look a little obliquely into the eye,
thus avoiding the light reflected from the lens, which gives bright
reflected images of the mirror.
When the eye is near-sighted, or myopic, the retina is beyond
the principal focus of the lens. This effect is produced in the
model by partly unscrewing the lens, taking care that it does not
fall out. An image of the retina is thus formed in front of the
198 OPTHALMOSCOPE.
lens, or the rays from it converge. Hence when employing the
direct method, a concave lens, as 1 or 2, must be placed behind the
mirror, to render the image visible. When the eye is far-sighted, or
hypermetropic, and incapable of viewing near objects, the retina is
nearer the lens than its focus. Hence the image is formed at a
considerable distance behind the lens, and can readily be viewed
by the direct method without any lens. This effect is imitated by
using the lens marked H, which has a longer focus than the other.
The difference is not perceptible by the indirect method, since it is
neutralized by a slight motion of the mirror.
Sometimes the cornea has a different curvature in horizontal and
vertical planes ; it is then said to be astigmatic. The third lens
marked A, shows this defect. With this, it is a little difficult to
view the retina clearly, since the focus is different in different
planes ; it will be noticed, however, that the papilla assumes an
elliptical, instead of a circular form.
Now replace the emmetropic lens, and insert the various diseased
retinas in turn, using the small diaphragm, or, if necessary, the
larger one. The retinas are numbered, and represent the following
conditions of the eye : —
1. Normal retina.
2. Atrophy of the papilla and retina.
3. Atrophy of the choroid.
4. Staphyloma posterior, an old case ; blood focus near the
macula lutea.
5. Hemorrhage of the retina.
6. Alteration of the retina.
7. Staphyloma posterior. Separation of the retina.
8. Infiltration of the papilla with blood.
9. Exudation of serous fluid, between the choroid and retina.
10. Glaucoma, with the circle of atrophy of the choroid around
the papilla.
11. Glaucoma and hemorrhage of the retina.
12. Atrophy of the papilla, and of the choroid around it.
The papilla is the point of entrance of the optic nerve. The
macula lutea, the point of the retina most used.
Atrophy means the gradual wasting away, and absorption of
any substance. Staphyloma, a thinning of the covering of the
INTERFERENCE OF LIGHT. 199
eyeball, especially around the optic nerve, allowing this portion
of the ball to extend backwards. Glaucoma is an increase in
quantity of the vitreous humor within the eye, causing a disten-
tion of the eyeball, accompanied with acute pain.
89. INTERFERENCE OF LIGHT.
Apparatus. To observe the interference of light, a diffraction
bank is employed, which consists of a long horizontal bar divided
into millimetres, and carrying sliding uprights, to which the fol-
lowing instruments may be attached, and placed at any desired
distance apart. A cylindrical lens to produce a bright line of
light, and a brass plate with a slit in it of variable width, like
that of a spectroscope. A biprism, or prism of glass with a very
obtuse angle, by means of which two closely adjacent images
of any object will be formed, and a double mirror designed for
the same purpose, whose two halves are inclined at a very small
angle, which may be varied by means of adjusting screws. To
observe the various effects produced, one of the uprights carries
a spider-line micrometer, or a simple eye-piece with cross hairs,
which may be moved laterally, and its position determined by a
millimetre scale and index. Or, this may be replaced by a small
direct vision spectroscope, to analyze any portion of the light
passing through the instrument. A screen of ground glass, or
paper, may also be substituted for the eye-piece. Although some
of the simpler phenomena are visible by ordinary light, yet to
obtain the best results sunlight is indispensible. An arrangement
is also desirable by which an intense monochromatic light may be
obtained, which may be done roughly by interposing colored
glasses, but much better by placing a prism in front of the slit,
throwing a ray of sunlight through it, and projecting the spectrum
thus obtained on the slit. When the day is cloudy, a soda or
lithium name may be employed. A cover should be placed 'over
the whole to cut off the stray light, or a simple piece of black
cloth may be employed for the same purpose.
Experiment. According to the Undulatory Theory all space is
supposed to be filled with a very rare medium, called ether, whose,
vibrations give rise to the phenomena of light. A luminous point
throws out concentric spherical waves, whose diameters increase
with very great velocity, and each of whose radii is called a ray
of light. The direction of the vibrations is transverse, that is, per-
pendicular to the ray, as is the case of waves of water, and the
terms crest and trough are here also used to denote the two
opposite positions of any portion of the ether. The distance from
200 INTERFERENCE OF LIGHT.
one crest to the next determines the color of the ray, and is called the
wave-length. In the same way, the intensity of the light depends
the height of the wave, or distance traversed by each particle. A
particle of ether can receive any number of systems of vibrations,
whatever their wave-length, intensity, direction, or plane of vibra-
tion, and will transmit each precisely as if the others did not exist.
If a particle receives two rays of light, precisely similar in every
respect, under the influence of both, its motion will be increased,
and a more intense light produced. Now suppose one of the rays
is retarded by half a wave-length ; its crests will coincide with, and
neutralize the troughs of the other ray ; accordingly the particle
will not move at all, and the result will be darkness. The same
effect will also be produced if one ray is retarded three, five, or in
fact any odd number of half wave-lengths, while if the retardation
is an even number of half wave-lengths, crest will fall upon crest,
and the light will be increased. This neutralization of one ray
by another, or light added to light, producing darkness, is called
interference, and by means of it many most important laws have
been established.
To produce interference, two precisely similar sources of light
are needed, at a very short distance apart. For this purpose,
place the cylindrical lens on a support at one end of the diffraction
bank, and throw a beam of sunlight through it. A very narrow
line of light is thus produced at its focus. Place the biprism on a
support, at a short distance in front of it, and two images will be
formed very near together, and precisely alike. If, now, a screen
is placed near the other end of the bank, its centre will appear
bright, since being equidistant from both images it will receive
simultaneously the crests and troughs of them both. If, how-
ever, a point is taken on one side of the centre, it will be nearer
one image than the other, and, accordingly, the crests and troughs
will not arrive simultaneously. If, then, the difference of path
is an odd number of half wave-lengths, darkness will be produced,
while an even number will give brightness.
The consequence will be a series of vertical black bands, corres-
ponding to 1, 3, 5, etc., half wave-lengths. As, however, the light
is white, and is composed of rays of all colors, and various wave-
lengths, the bright and dark spaces will be at different distances
INTERFERENCE OF LIGHT. 201
for each, consequently a series of colored bands will be produced,
with a white centre. These bands are much more visible with the
eye-piece, and their number is increased by employing monochro-
matic lights.
Their position affords a means of determining approximately, the
length of a wave of light, as follows. Bring the cross-hairs to
coincide successively, with each of the visible bright bands, when
monochromatic light is used, and read the position of the eye-
piece. Measure, also, the distance from the slit, or focus of the
cylindrical lens to the cross-hair. If, now, the distance between
the two images can be determined, a simple calculation will give
the difference in their distances from the bright band, that is, one,
two or three times the wave-length, according to the number of
the band. The distance between the images may be found by the
following device. Insert a lens between them and the eye-piece,
having a focal length about one fourth their distance apart.
There will be two positions, in which the images will be distinctly
seen. Bring the cross hairs to coincide with their images, as
formed with the lens, by moving it laterally, taking care to move
the lens only, when focussing. In one case, the distance between
the two luminous lines will be magnified, and in the other, dimin-
ished, in precisely the same ratio. Accordingly, the mean pro-
portional of these two measurements will give the true distance
with accuracy.
To calculate the wave-lengths from these measurements, let D
be the distance from the slit to the cross hairs, d the distance
apart of the two images, and b the distance of the first band from
the centre, equals one half the distance between the images on op-
posite sides of the centre line. Then by similar triangles, since
b and d are always very small, compared with D, it follows that
D : ~b = d: l the required wave-length. In the same way the
other bands give 2x, 3A, etc. Repeat this measurement, with
other positions of the eye-piece, and rays of other colors, and
notice that the more refrangible the ray, the shorter the wave-
length.
As the position of the eye-piece is varied, any given band will
evidently lie on a hyperbola with the two images as foci, since it
is the locus of a point, whose distances from two others differs by
202 DIFFRACTION.
a constant amount. To prove this, observe the position of one of
the outer bands, varying D five centimetres at a time, and repre-
sent the results by a curve, in which D gives the abscissas, and
an enlarged value of 5, the ordinates. Construct also the theoret-
ical curve on the same sheet.
Now replace the eye-piece by the spectroscope, and moving it
laterally, the bright and dark bands, in turn, fall on the slit, and
the colors of which they are composed may then be determined
with precision. Thus starting with the central bright band, it will
be found to give a continuous spectrum, traversed, of course, by
the usual solar lines. In the first black band, all the colors disap-
pear, then all the colors reappear, beginning with the violet, and
as the slit is moved still further, a dark band will enter the violet
end of the spectrum, and will traverse it, soon to be followed by a
succession of others, whose distance apart becomes less and less,
until finally, many are visible in the spectrum at the same time.
Similar effects may also be obtained with the double mirror,
taking care that the two halves are slightly inclined, and that
their edges meet exactly. This is accomplished by means of the
adjusting screws. It will then form two closely adjacent images
of any object reflected in it. Place the mirror in such a position,
that it shall reflect two images of the slit (which is moved a
short distance to one side) along the bank, when the bands are
formed by interference, precisely as with the biprism. A dia-
phragm must be interposed, to cut off the direct rays of the slit
from the eye-piece. As the interval between the two images
diminishes, the bands become more spread out, as may be shown
by diminishing the inclination of the two halves of the mirror,
by means of the adjusting screws.
90. DIFFRACTION.
Apparatus. Besides the diffraction bank employed in Experi-
ment 89, a number of brass plates are needed, which may be
inserted in the sliding uprights, and which are perforated with
apertures of various shapes and sizes. Thus one will carry a slit
of adjustable width, a second a large aperture half covered by a
plate with a vertical edge, others with two closely adjacent slits,
a vertical wire, circular holes of various sizes, and some with two
holes a short distance apart. An immense variety of effects may
be obtained by using apertures of different shapes, and sometimes
DIFFRACTION. 203
a number of these are photographed on a plate of glass and
brought successively into the axis of the instrument. A plate
of glass on which a large number of equidistant fine lines are
ruled is also needed, and pieces of wire gauze and lace with
square and hexagonal meshes. The effect of these various ap-
ertures is best shown by placing them in front of a telescope
directed toward an artificial star, as in Experiment 84, or the
optical circle may be used, replacing the slit by a minute circular
aperture.
Experiment. The phenomena of diffraction are best explained
by a comparison with the similar effects produced by water.
Suppose a long straight wave, like a breaker rolling in on a
beach, encounters in its passage a rock, or the end of a break-
water. After passing, instead of leaving the water quite at rest
behind the obstacle, the end of the wave will spread inwards, so
that if the rock is small the two portions will meet after moving
some distance, and no portion of the surface not close to the rock
will remain perfectly level. When the undulatory theory was
first proposed, it was claimed by its opponents that the waves of
light would, in like manner, pass around an obstacle, so that
shadows could not exist, and again that a beam of light could
have no definite edges, since it would spread out on all sides like a
wave of water after entering the narrow inlet to a bay. In point
of fact, this spreading actually takes place, but as each wave is
followed by millions of others similar to it, interference takes place
so that but little remains, except in the direction of the beam.
In fact, it is only by taking special precautions that the remaining
rays can be detected, and the phenomena then observed are
known by the name of diffraction.
Place the cylindrical lens, or the slit, at one end of the diffraction
bank, and throw a beam of sunlight through it. Place a screen at
the other end, and between them the plate with the aperture half
closed by a piece of sheet brass having a vertical edge, or a slit
with one side removed. The shadow will now be cast upon the
latter, and examining its edge closely, it will be found that a small
amount of light has been bent inwards or inflected. Outside of
the edge, and parallel to it, a series of colored bands or diffraction
fringes appear, due to the partial interference of these rays. They
are seen on a much larger scale by means of the eye-piece, and
204
DIFFRACTION.
their constitution is well shown by means of the direct-vision
spectroscope, as is the case of the interference fringes.
Now replace the other side of the slit, and as it is gradually
narrowed new fringes will appear in the shadow, which finally will
quite obscure the others, leaving an appearance very like that pro-
duced by interference with a biprism. When the slit is moder-
ately narrow, both system of fringes are visible, those in the
interior being either, bright or dark-centered, according to the dis-
tance of the screen. Next use a narrow screen, as a wire, instead
of the slit, in the middle of the bank, when a series of fringes will
be obtained, both inside and outside the shadow, and varying
their distance apart with the diameter of the wire. By using two
slits near together, fringes are also produced by interference as
with the biprism. Strangely enough, these efforts are quite in-
dependent of the material of the slit, its thickness, or physical state.
After verifying the above facts, measure the form of some of
the fringes, as in the case of interference, and see if their relative
distances from the centre agree with theory.
When rays from a real or artificial star pass through an
aperture, those striking the edges are diffracted, so that they
are thrown off obliquely in directions dependent on their wave-
lengths and the form of the aperture. I£ therefore, they are
received in a telescope, the direct rays will form a bright spot,
or image of the star, which will be surrounded with colored
fringes or bands of very various forms.
To begin with the simplest case, suppose the aperture circular
and of considerable size, as in a common telescope. The dif-
fraction will be very slight, but quite perceptible, with a good
instrument and a high power, although with a poor lens it is some-
times obscured by the aberration. The true angular diameter
of the fixed stars is so exceedingly small that it would be quite im-
possible to observe their disks with any power yet employed.
With a good telescope, however, small bright circles are seen,
called spurious disks, which increase in diameter as the aperture
is diminished, and which are surrounded by one or more colored
rings, due also to diffraction. If, now, a triangular aperture is
interposed in front of the objective, the star is seen to be sur-
rounded with six diverging rays, corresponding to the angles and
WAVE LENGTHS. 205
centres of the sides of the triangle, while a square aperture gives
a star with four rays. Try, in the same way, the other apertures
and the gauze, with which a vast variety of curious effects may be
obtained. On interposing the series of equidistant lines, a number
of colored bands are seen on each side of the star, with their
violet ends towards the centre, and their direction perpendicular
to the lines on the glass. This effect is most important, as it
affords a means of determining the length of a ray of light with
the utmost precision, as will be described in the next experiment.
In all cases the distances of the colored rays will depend on
their wave-lengths, being greatest for the red and yellow, then for
green, and least for the blue and violet. This may be shown by
employing monochromatic light, which may be obtained by illumi-
nating the slit or artificial star by different parts of the spectrum,
formed by allowing a ray of sunlight to pass through a prism, or
more simply by interposing colored glass, or employing a soda
flame.
Many familiar phenomena are due to d infraction ; for example,
the halo around a distant light seen through a window covered
with moisture or frost, or the same effect produced on the -sun or
moon by fog. When the particles are all of nearly the same size,
colors become visible, which may be distinguished from the ordi-
nary large halos, both by their small size, and by noticing that in
halos produced by diffraction the red is always outside, while in
halos caused by refraction in minute crystals of ice, the red is
always inside, since this color has a smaller index of refraction,
but greater wave-length. This effect may be produced artifi-
cially by scattering on a plate of glass, lycopodium, or any fine
powder whose particles are all of nearly the same size.
91. WAVE LENGTHS.
Apparatus. The optical circle, and a glass plate, on which are
ruled several thousand, very fine, equidistant lines, at intervals of
about one hundredth of a millimetre. Sunlight is desirable for
this Experiment, but if the day is cloudy, a soda and lithium
flame may be employed instead.
Experiment. One of the most important applications of diffrac-
tion, is to the measurement of the lengths of waves of light of
206 WAVE LENGTHS.
various colors. The fringes obtained with screens traversed by
very fine lines, or gratings, as they are called, are employed for this
purpose, and give very accurate results.
Sejt the two telescopes of the optical circle opposite each other,
adjust them for parallel rays, and place the glass plate on the
centre stand between them, and at right angles to their axes.
Reflect a ray of sunlight through the slit of the collimator by
means of the mirror, and on looking through the observing tele-
scope, the following appearances will be visible as it is moved from
side to side. In the centre will be a brilliant white image of the
slit, and on each side spectra will be seen, with their violet
ends turned towards it. The first will be bright and short,
and each in turn fainter and longer, until finally they over-
lap, forming a continuous band of light. Focus with care, and
nearly close the slit, when the solar lines will become visible in
each spectrum, as in a spectroscope, except that in the present
case the red end is much more extended, the violet more crowded
together. Care must be taken that the lines are vertical, as the
spectra being perpendicular to them will otherwise fall out of the
field, above on one side, and below on the other. The plate should
also be perpendicular to the axis of the collimator, or an error
will be introduced in the angular distance of the spectra.
The formation of these spectra, is explained as follows. Let M^
-ZVJ 0, Fig. 64, represent the rays from the slit, after being rendered
parallel by the lens of the collima-
tor, so that they shall all be in the
same phase, or state of vibration,
when they strike the plate of glass.
The lines on the latter, are shown
on a greatly enlarged scale, at A,
13) C, J), which represent a section
at right angles to their length.
Being opaque, or at least only
translucent, they divide the sur-
face of the glass, into a number of equidistant narrow apertures,
through which the light passes. As was proved by Huyghens,
each of these may be regarded as a new source of light, from
WAVE LENGTHS. 207
which the rays pass out in all directions, and in general, interfere
and neutralize each other. There are, however, certain directions,
aaAP,m which the difference of path of AP and BQ, equals
exactly one wave-length /*, and in this case they will unite,
and light will be produced. The ray CR will add its effect,
since it differs by exactly two wave-lengths, and in the same
way, all the other rays from the other apertures unite and pro-
duce a bright light in the direction AP. The direction of PR
is that of the front of the wave, or perpendicular to AP, and
drawing AF parallel to it, the condition that light may be pro-
duced is evidently that in the right-angled triangle AFB, FJB
shall equal L Call d the distance between the lines, in frac-
tions of a millimetre, and a the angle the ray AP makes with the
axis of the collimator, equals FAS. Then X = d sin a, from
which, knowing d and a, A may be computed. To find a, bring
the cross-hairs of the observing telescope successively to coincide
with the image of the slit and the given ray in the first spectrum,
and the difference in the readings of the vernier gives the required
angle; or better, read the position of the correspondh.g rays in
the spectra on the right and left of the central image, and divide
the difference by two. Make a similar observation with three or
four of the prominent lines. Suppose, now, that a is so much i
increased that AF = 2A ; evidently light is again produced,
which gives rise to the second spectrum on each side. The third
and fourth spectra are accounted for in the same way. Measure
the position of the lines before observed in all of them, and com-
pute X from each, taking care to divide by two for the second, by
three for the third, etc., to get the true wave-length. The mean
of these observations should then agree closely with that given on
page 152. If d is not given, it should be determined on the
dividing engine, or with the microscope and spider-line microm-
eter.
If the lines on the glass plate are well ruled, very beautiful
spectra may be obtained, in some cases almost equal to those
formed by the best spectroscopes. Generally the spectra on one
side are better than those on the other, probably owing to some
want of symmetry in the two sides of the lines. Again, often
one of the spectra will be fainter than the next beyond it, or even
208 POLARIZED LIGHT.
wanting altogether, it may be proved analytically that the mih
spectrum will be wanting, when the ratio of the width of the
lines to the spaces between them, is as n : n', and m = n -f- n'.
That is, if the dark spaces are half as broad as the bright, n — 1,
n' = 2, m = 3, and the third spectrum will be wanting.
92. POLARIZED LIGHT.
Apparatus. A rhomb of Iceland spar, and examples of the five
methods of polarizing light, that is, by reflection, by refraction or
by a bundle of plates set at an angle of 55°, and by the three
methods of double refraction. These consist of a double-image
prism, a Nicol's prism, and a tourmaline plate. Fig. 65 represents
a form of polariscope which will be found both simple and effective.
jB is a plate of glass resting on a piece of black velvet, A
a screen of ground glass, and D a Nicol's prism. This is so
placed that the angle of incidence of the ray reflected from the
centre of B shall be 55°, and consequently shall be totally polar-
ized. C is a plate of glass on which the object to be examined is
laid. Various objects should accompany this instrument, as a
plate of selenite, some figures made of the same material, some
pieces of uriannealed glass, and two small screw-presses by which
small squares or rods of glass may be subjected to longitudinal
or transverse strain. Also some lenses, glass stoppers, and other
articles imperfectly annealed, and some spectacle lenses of quartz
and glass. Plates of the following series of crystals should be
provided. 1. Iceland spar cut perpendicular to the axis; 2,
quartz ; 3, arragonite ; 4, topaz ; 5, borax ; 6, nitre ; 7, double plate
of quartz giving hyperbolas ; 8, Savart's bands. This list may be
greatly extended, and it is well to add any novelties that can be
found, with a written description appended. All these objects
will appear to much greater advantage if the outside light is cut
off, either by a black cloth, or by a cover fitting over the polari-
scope, and extending from A to D in the figure.
Experiment. According to the Undulatory Theory, light is
produced by vibrations of the ether at right angles to the direc-
tion of the ray. If, then, the latter moves vertically, all the mo-
tions will be horizontal, and in common light, some north and
south, others east and west, and others in various intermediate
directions, that is, in all planes passing through the ray. If, now,
the vibrations can in any way be confined to a single plane, the
light is said to be . polarized, and this plane is called the plane of
vibration of the ray. A plane perpendicular to this and passing
POLARIZED LIGHT. 209
through the ray, is called the plane of polarization. It would
be much better to have taken the latter plane as coincident with
the former, but, unfortunately, the name was given before the
direction of the vibrations was known.
Although it is impossible to detect by the eye alone whether
light is polarized or not, yet many substances affect it differently,
according to the direction its plane bears to some line in them, so
that when it emerges from them, it no longer possesses the prop-
erties of plane polarized light. To examine the effect produced,
the ray is first passed through the polarizer, as it is called, by which
all its vibrations are brought into one plane, it is then allowed to
pass through the substance under examination, and finally tested
by the analyzer, which may be made precisely like the polarizer,
and is used to detect any change effected in the ray.
There are five forms of polarizers in common use. First, by re-
flection. When a ray of polarized light impinges on a plane
surface of a transparent medium, the amount reflected depends on
the direction of the plane of polarization, and the angle of inci-
dence. If the plane is perpendicular to the plane of incidence,
and the angle is such that the reflected and refracted rays shall be
perpendicular, all the light is transmitted. The angle of incidence
is then called the angle of total polarization, and its value may be
determined as follows. Let n be the index of refraction of the
medium, i the angle of incidence, which equals the angle of reflec-
tion, and r the angle of refraction. Then, since the reflected and
refracted rays are at right angles, * + r — 90°, but sin i = n sin
r = n sin (90 - i) = n cos r, and tang i = n. Common light
being composed of rays polarized in every plane passing through
the beam, may be regarded as composed of two equal rays, polar-
ized at right angles, just as all forces acting on a point in a plane
may be divided into two components at right angles to each other.
Regarding, then, the light as composed of two beams, one A, polar-
ized in the plane of incidence, and the other, .2?, polarized at right
angles to it, evidently none of the latter will be reflected ; hence
the reflected ray will be entirely composed of light polarized in the
plane of incidence, or will be totally polarized. The value of
* is about 55° for glass, and 53° for water. The simplest form of
polarizer is therefore a plate of glass, on which the light impinges
14
210 POLARIZED LIGHT.
at an angle of 55°. Commonly the lower surface is blackened, or
black glass is employed, but there is no advantage in this, in fact
it is better to use several plates of clear glass, to increase the
light.
As a portion of A is turned back by the glass, evidently the
refracted beam will be partially polarized, being composed of the
whole of -Z?, and part of A. By using a number of plates, each
will reflect a portion of -4, leaving B unaffected ; the latter may
thus be almost completely freed from A> or the light nearly
perfectly polarized.
The other three polarizers depend on double refraction. If a
ray of light is allowed to pass through any crystal not of the
monometric system, it will be divided into two parts, one called
the ordinary ray, which will follow the usual laws of refraction,
and the other, the extraordinary ray, which will follow new laws.
To show this, lay a crystal of Iceland spar on a piece of paper on
which is marked a single dot. The latter will now appear double,
and if the crystal is turned, one image, the extraordinary, will
revolve around the other. These two rays are found to be polar-
ized in planes at right angles to each other, but in the present
case are not sufficiently separated to be conveniently employed.
They, moreover, emerge parallel, that is, they are no more sep-
arated for distant objects than for near, since the plate being
bounded by parallel faces, the second surface neutralizes the angu-
lar divergence produced by the first. To remedy this defect,
prisms of glass and spar are cemented together in such a way
that the refraction of one ray shall be compensated, while the
other will pass out obliquely, giving two images separated by an
angular amount of two or three degrees. This combination is
called a double-image prism.
Another arrangement is the Nicol's prism, which consists of a
rhomb of Iceland spar cut diagonally, and the two parts cemented
together again with Canada balsam. This substance has an index
of refraction greater than the extraordinary, but less than the
ordinary ray in spar, consequently the former will pass through
unchanged, while the latter being totally reflected will be thrown
out on one side, and will be absorbed by the black paint covering
the prism. The light passing through will therefore be polarized
POLARIZED LIGHT. 2ll
in a plane passing through the ray and the longer diagonal of the
rhombus at the end of the prism. This is called the principal
plane, or simply the plane, of the prism.
The fifth form of polarizer is a plate of tourmaline, cut parallel
to the axis, which posseses the curious property of absorbing the
ordinary ray, so that the emergent light is polarized in a plane
parallel to its axis, or greater diameter.
Either of these instruments may be employed as a .polarizer,
but each has its special advantages and defects. The method of
reflection is the simplest, and a beam of any size, perfectly polar-
ized may be obtained by it, but there is much loss of light from
the transmitted rays, and the change of direction is often an
objection. The bundle of glass plates give a large beam, but the
polarization is not very perfect unless a large number of plates is
employed, and then the loss by absorption is considerable. By
the other methods very large beams cannot be obtained. The
double image prism gives excellent results when the presence of
the second beam is not objectionable, or when, as sometimes
happens, it can be thrown out to one side of the apparatus. The
Nicol's prism is more employed than any other polarizer, but when
of large size it is very expensive. A tourmaline plate is also good,
but if very thin the ordinary ray is not wholly absorbed, and the
polarization is not complete ; while if thick, the ray is strongly
colored. Colorless tourmalines exist, but unfortunately are not
opaque to the ordinary ray, and hence do not polarize the trans-
mitted light. Examples of these various polarizers will be found
on the table.
When a ray of polarized light is viewed through a Nicol's prism,
or other polarizer, the amount of light transmitted varies as the
prism is turned. Thus allow a ray of light to pass through a
Nicol's prism with its principal plane vertical, that is, so that the
transmitted light shall be polarized in a vertical plane. If this
beam is viewed with a second Nicol's prism, it will be found that
as the latter is turned, the amount of light transmitted varies,
being greatest when its plane is vertical, and nothing or all the
light cut off, when the plane is horizontal. This evidently follows,
since the prism then transmits only light polarized vertically. In
intermediate positions, the amount of light is determined by
212
POLARIZED LIGHT.
decomposing the ray into two at right-angles, as in the case of
forces, only it will be proportional not to the cosine, but to the
square of the cosine of the angles. Try the other polarizer, in the
same way, and it will be found in all cases, that when their planes
are parallel, light is transmitted, but when turned at right-angles,
or crossed, as it is called, the light is cut off. Accordingly, to
test for polarized light, view the beam through a Nicol's prism or
tourmaline, which is then called an analyzer. If there is no change
of brightness of the transmitted ray, the light is unpolarized, while
if in certain positions all the light is cut off, the polarization is
complete. Next, turning the analyzer, find the position in which
the field is darkest, when its plane will be perpendicular to the
plane of polarization.
To apply this to some familiar objects, examine the light re-
flected from the top of a varnished table, and it will be found to
be strongly polarized in a vertical plane. Moreover, when this
light is cut off, the color of the wood and its grain is much better
seen. It has been proposed to use Nicol's prisms in this way for
viewing oil pantings, thus cutting oif the troublesome reflection.
Sometimes the light reflected from the front of glass cases renders
it difficult to distinguish objects within them. A Nicol's prism is
then often very serviceable. Again, it has been proposed to use
a Nicol's prism to cut off the light reflected by water, to render
rocks or other objects beneath its surface more visible. To show
this effect, place a coin at the bottom of a vessel of water, or
under several plates of glass, and allow a strong light to fall on it.
It may then be easily seen when the polarized light is cut off,
although otherwise quite invisible. As another example, view the
two dots seen through a crystal of Iceland spar, and they will be
found polarized in planes at right-angles. If the two images are
connected by a line, it will lie in the plane of the ordinary image,
or fixed dot, around which the other appears to revolve.
To view any body by polarized light, the instrument represented
in Fig. 65, will be found both simple and effective. B is the
polarizer, consisting of one or more plates of glass, and D a Nicol's
prism, serving as an analyzer which may be turned by any desired
amount, and which is set at such an angle that the light reflected
from the centre of B shall be totally polarized ; C is a plate of
POLARIZED LIGHT. 213
glass on which the object to be examined may be laid, and A a piece
of ground glass, to cut off the reflection of outside objects, and to
render the field of view bright and
uniform. The light reflected from
B is polarized vertically ; accord-
ingly, when D has its plane verti.-
cal, the field is bright, when hori-
zontal, the field is dark.
Suppose now, any doubly re-
fracting medium is inserted be-
tween the analyzer and polarizer;
for instance, a plate of selenite Fig> ^
laid upon C. The ray on entering
the selenite is divided into two, the ordinary and extraor-
dinary, polarized at right-angles, the plane of the ordinary
passing through the axis of the crystal. The relative intensities
of the two will depend on the position of this axis with regard to
the plane of polarization of the ray, and may always be obtained
by decomposing the latter into two parts; one the ordinary ray,
coinciding with the axis, the other at right angles to it. If the axis
is perpendicular to the plane of polarization, evidently all the
light will pass into the extraordinary ray, while if they coincide
all becomes ordinary. The two intensities are equal when the
angle between the axis and plane is 45°. Now the two rays travel
through the crystal with unequal velocities, as is shown by their
different indices of refraction. On emerging, one ray will be
behind the other by an amount dependent on the thickness of the
plate ; for instance, one half wave-length of yellow light. The
two rays, however, cannot interfere, since they are polarized, and
are therefore vibrating, at right-angles. Therefore the crystal
will still appear colorless to the eye. If, however, it is viewed with
a Nicol's prism with its plane vertical, the two rays are again
decomposed, the horizontal components cut off, and the vertical
portions brought together so that they can interfere. If, then,
white light is employed, which is composed of rays of all colors,
the yellow portion will be stricken out, and the remainder will be
of the complementary color, or purple. Now turn the analyzer
90°. The rays before cut off will now be transmitted, and vice
214 POLARIZED LIGHT.
versa, accordingly the color of the light will be yellow. As the
analyzer is turned from these points the colors become fainter and
fainter, until at the 45° points all the rays are equally affected, and
the light becomes white.
In the same way, on turning the selenite plate, the two compo-
nents are equal only when the a^ds is inclined at an angle of 45°,
in which case the interference is complete. In other positions one
component is larger than the other, the interference is only partial,
and the colors are fainter, a part of the light being white. When
the angle becomes 0°, or the axis lies in the plane of polarization,
all of the light passes into the ordinary ray, none into the extra-
ordinary, and consequently there is no interference, and the light
is white. ' In the same way, when the axis is perpendicular to the
plane of polarization, all the light enters the extraordinary ray,
and the result is again white light. Accordingly as the selenite is
turned, the color becomes fainter and fainter, and disappears at
the 0° and 90° points, but on passing them does not assume its
complementary tint.
By varying the thickness of the plate, the amount of retard-
ation may be varied at will, and with it the wave-length of the
ray stricken out, and the color produced. Figures are sometimes
made of selenite to represent birds or flowers, each portions hav-
ing such a thickness that when viewed by polarized light they will
assume their proper colors. They are then mounted in Canada
balsam between two plates of glass, and by ordinary light being
transparent are almost invisible. Placing them on GY, however,
they appear in gorgeous colors, which disappear as the analyzer D
is turned, and again reappear in complementary tints, as the rota-
tion is continued. If, however, the selenite is turned, the colors
fade, but reappear unchanged.
All transparent bodies will produce double refraction, and affect
polarized light when subjected to unequal strains in different
directions. In the cases mentioned above, this is effected by
crystallization, but it may also be produced mechanically. To
show this, place a square of glass in the small press, and lay it on
G7, turning D so as to cut off the light. No effect is now pro-
duced, but as soon as the glass is compressed by turning the screw,
bright spaces appear at the points where the pressure is greatest,
POLARIZED LIGHT. 215
and which as the screw is turned, increase in size, and finally
become colored. Care must be taken not to exert too great a
pressure or the glass may be fractured. Apply in the same way
a transverse strain to a rod of glass with the other press, and
sketch the appearance. Still more care is needed in this case not
to break the glass.
When glass is cooled suddenly the exterior contracts, and when
the interior cools, the whole is subjected to great strain, rendering
it very brittle. To remedy this, glass vessels intended for com-
mon use are annealed, that is, heated arid allowed to cool very
slowly. Place the pieces of unannealed glass on (7, and very
curious and beautiful markings will appear, which vary with the
form of the specimen, and the position of the analyzer. Common
glass objects, as stoppers to bottles, may be tested in the same way,
to see if they have been properly annealed. The best practical
application made of this principle, is to test lenses, as already
mentioned in Experiment 84.
When a ray of light passes through a crystal, not of the mono-
metric system, the effect produced varies with the direction. In
the dimetric and hexagonal systems, when the ray passes through
the crystal in the direction of its principal axis, it is not divided
into two, or more properly, the two follow the same path but with
different velocities. This direction is called the optic axis of the
crystal, and such crystals are called uniaxial. Now place a rhomb
of Iceland spar with its principal axis vertical, that is, so that the
corner formed by the angles of 120° shall be uppermost, and the
three adjacent faces equally inclined to the vertical. Now, as in
the case of crystallographic axes, not only the line through the
centre of the crystal, but any vertical line will be an optic axis.
The principal section of any plane of this crystal is a vertical
plane perpendicular to it. If the incident ray lies in a principal
section, the extraordinary ray will lie in the plane of incidence,
otherwise to one side of it. Crystals of the trimetric, monoclinic,
and triclinic system, have two optic axes which may be inclined
at any angle with each other. Such crystals are called biaxial.
When light slightly inclined to the optic axis passes through
the crystal, interference takes place, producing brightness or dark-
ness according to the amount of retardation, or angle of inclina-
216 POLARIZED LIGHT.
tion. If rays are allowed to pass in all directions through the
crystal, the optic axes will be seen to be surrounded with circles
alternately bright and dark, and colored, owing to the unequal
wave-lengths of the different rays. To observe them, it will not
do to lay the crystal on (7, as the rays would then all be nearly
parallel, but it must be held close to 7>, and inclined from side to
side. Object No. 1 is a crystal of Iceland spar, cut perpendicular
to the axis, and gives readily the series of rings. A back cross is
also formed with its centre in the axis which changes to white
when the analyzer is turned 90°. In observing all these crystals, it
will be noticed that the rings change only when they are inclined,
and not when moved parallel to themselves, showing that the optic
axis, as stated above, has no particular position, but is a certain
direction. A common method of observing the rings is by
tourmaline tongs, or two plates of tourmaline with the crystal
placed between them, and the nearer one, or analyzer free to turn.
Finer effects may be obtained by lenses forming a sort of micro-
scope, but this arrangement is less simple than the above. Object
No. 2 is a plate of quartz cut in the same manner. Quite a differ-
ent effect is here produced, partly because the retardation is much
less, unless the plate is very thick, and hence the rings much
more widely spread, and partly because quartz produces what
is called rotary polarization, that is, it twists the plane of polariza-
tion from its original position by an amount depending on the color,
and proportional to the thickness. Accordingly the plate will ap-
pear colored, the tint varying as the analyzer is turned. Next try
some biaxial crystals. No. 3 is specimen of arragonite, in which
the two axes are visible, surrounded with colored rings, and with
a double cross passing through them, which changes into a hyper-
bola as the crystal is turned. No. 4 and 5 are crystals of topaz
and borax, in which the axes are so far separated that only one
can be seen at a time. The other may sometimes be found by
inclining the crystal. No. 6 is a crystal of nitre, in which the axes
are only separated about 5°, and hence are both easily seen
together. The separation of the rings depends on the thickness
of the plate, and the difference of the ordinary and extraordinary
indices of refraction, and is therefore quite independent of the
axes. Some crystals give peculiar systems of rings which vary
POLARISCOPE. 217
with each different specimen. Curious effects may be obtained by
combining two or more plates in various ways. The most impor-
tant are the following. No. 7, two plates of quartz cut parallel to
the axis, turned at right angles, and then cemented together. A
system of equilateral hyperbolas is thus obtained with a common
centre. No. 8 is formed of two plates cut at an angle of 45° with
the axis, and crossed in the same way. They give a series of
rectilinear bands, forming in fact the ends of the hyperbolas of No.
7 with their asymptote, to which they are now parallel. This com-
bination is known as Savart's plate, and is important, as forming
one of the most delicate tests for polarized light. The centre band
may be rendered either white or black by turning the analyzer
90°. They are most intense, when either parallel or perpendicular
to the planes of both analyzer and polarizer.
93. POLAKISCOPE.
Apparatus. A stand is employed like a theodolite, or altitude
and azimuth instrument, only the circles need be divided no finer
than to degrees. The various forms of polariscopes and polari-
meters described below, may be attached to this, so that they are
free to turn around their axes, the angle of rotation being meas-
ured by a graduated circle and index. When the object to be
examined is very minute a telescope is needed, with a positive
eye-piece, in which is a Nicol's prism. In front of this, a slide is
placed, by means of which a biquartz or Babinet's wedges may
be interposed at the focus. The circle is attached to the eye-piece,
and acts like an eye-piece goniometer (p. 163). For common
objects, the telescope is replaced either by a Nicol's prism, in front
of which a Savart's plate may be inserted, or by an Arago's
polariscope.
Two forms of polarimeter are employed, the first or common
form, proposed by Arago, consisting of a Savart's polariscope, in
front of which one or more plates of glass may be inserted, and
turned at any desired angle so that the light may be more or less
strongly polarized by refraction. A graduated circle serves to
determine the angle through which they are turned. The second
form of polarimeter consists of an Arago's polariscope, in which
the selenite plate is removed, and a Nicol's prism with a graduated
circle placed in front of the double-image prism, so that it may
be turned through any desired angle with regard to the latter.
These may also be mounted *on the stand like the polariscopes, so
that they may be pointed in any desired direction.
218 POLARISCOPE.
Experiment. The simplest method of detecting the presence
of polarized light is by a Nicol's prism, or other polarizer, as
described in the last Experiment. Examine in this way the light
reflected from the surface of the table, from a glass plate and
other sources. Turn the Nicol until the least light is transmitted,
and the direction of the shorter diagonal of the face of the prism
will give the plane of polarization. This method, however, is not
very sensitive, as the variation in intensity of the light is not
perceptible, unless a considerable proportion is polarized. A more
delicate instrument is the Arago polariscope. This consists of a
tube with a square aperture at one end, and a double-image prism
and plate of selenite at the other. The size of the aperture is
such, that the two images of it shall be just in contact, but not
overlapping. If, now, a ray of polarized light is viewed through
the prism, the two images will in general assume complementary
colors. When the line of separation of the two images is
parallel or perpendicular to the plane of polarization, the colors
are most strongly marked, and they disappear when the angle of
inclination is 45°. To determine the plane of polarization of any
ray, first direct the instrument towards the light reflected from
a polished horizontal surface, turn the line of separation of the
two images until it is vertical, or parallel to the plane of polar-
ization, and note the color of the right hand image. Now direct
the tube towards the source of light and turn it until this image
has the same color as before. The plane of polarization is then
parallel to the line of separation. It is well to make a notch in
one side of the square, which will then appear in a different part
of the two images and thus serve to distinguish them. The exact
direction of the plane of polarization may be found by noting
when the colors are most marked, or, more accurately, by bisect-
ing the two positions where they disappear, each of which is 45°
distant from it. The delicacy of thi§ instrument is much greater
than that of a simple Nicol's prism, as with it about three or four
per cent, of polarized light can be detected.
Sometimes the Arago polariscope is used without the tube.
For instance, in observing the polarization of the solar corona
during a total eclipse, doubt was cast on the results by the strong
polarization of the sky. To eliminate this, the tube was removed,
POLARISCOPE. 219
in which ease the two images of the sky overlapped, producing
unpolarized light, while the images of the corona were separated
so that they appeared on a white unpolarized back-ground.
A still more delicate form of polariscope is that proposed by
Savart, which consists of a Nicol's piism with a double plate of
quartz, giving bands as described in Experiment 92, specimen No.
8. The plate is attached to the Nicol so that the bands shall be
perpendicular to its principal plane, in which case, when parallel
to the plane of polarization they will be black-centred, and when
perpendicular to it, white-centred. If the bands were parallel to
the plane of the Nicol, this effect would be reversed. It is now
very easy to determine the plane of polarization of a given ray.
The instrument is turned until the bands are black-centred, when
their direction marks that of the plane. The position of the latter
is then found more precisely by bisecting the two points of disap-
pearance of the bands. This instrument is more sensitive than
either of the preceding, as by it one or two per cent, of polarized
light can be detected. Try these different instruments on various
sources of polarized light, and see if all give the same results for
the direction of the plane of polarization. For instance, see if the
light reflected by paper, wood or cloth is polarized in the plane
of incidence, and if that transmitted obliquely through glass is
polarized in a plane perpendicular to the plane of refraction.
Now direct the telescope towards some source of polarized light,
and observe its plane with the simple Nicol's prism. Then push
the slide so as to interpose the biquartz, which consists of two
pieces of quartz joined together, one turning the ray to the right,
the other to the left. The two halves will then assume comple-
mentary colors, unless the plane of polarization is parallel or per-
pendicular to their line of junction. In the first case, the color
of both is a sort of pale violet, in the second, yellowish brown.
Make a number of observations of the angle of the plane, and
compute the probable error. Now try the effect of the other
quartz plate. This is composed of two wedges of quartz cemented
together, one turning the ray to the right; the other to the left.
Accordingly a series of bands are produced, which disappear when
parallel or perpendicular to the plane of polarization. Repeat the
observations with this plate, and compare its probable error with
220 POLARISCOPE.
that of the biquartz. It will be noticed that both these devices
require a parallel beam, while the Savart's polariscope, which needs
a converging beam, cannot be attached to a telescope in this way,
but must be placed in front of the eye-piece. In this case it cuts
down the field of view, and is therefore inconvenient to use.
To measure the proportion of polarized light in a given beam,
the polarimeter is employed. This consists of a Savart, or other
form of polariscope, in front of which some plates of glass are placed,
free to turn, so that the transmitted light may pass through
them at any desired angle. It will thus be polarized by refrac-
tion to a greater or less extent, depending on the number of plates,
and the angle through which they are turned. To measure this,
a graduated circle is attached, which may be divided either into
degrees, or so as to give the percentage directly. The bands
should be placed parallel to the axis around which the plates turn.
To use this instrument, set the plates at 0° and direct it towards a
source of unpolarized light. The field will now be perfectly uniform.
Turn the plates, and the bands will appear faintly, dark-centred
and increasing in strength with the angle. Turn the plates back
to 0°, and direct the instrument towards the light to be examined,
find its plane of polarization and bring the bands to a position at
right angles to it, that is, so that they shall be most strongly light-
centred. ISTow on turning the plates, they tend to neutralize the
polarization, since they tend to polarize it in a plane passing
through this axis, while it is already polarized in a plane perpen-
dicular to this. As they are turned, the bands therefore become
fainter and fainter, then disappear and reappear dark-centred,
when the angle becomes too great. At the point of disappearance,
the polarization produced by the plates is just equal to that already
present in the beam, the transmitted light is therefore unpolarized,
and gives no bands. Take a number of readings of the point of
disappearance, first turning the plates to the right, and then to the
left, and reduce to percentages by means of a table which should
accompany the instrument.
The difficulty of computing this table with accuracy, greatly
diminishes the value of this instrument. The theoretical formulas
are quite complex, and of little use on account of the difficulty of
allowing for the light absorbed by the glass. It must therefore be
POLARISCOPE. 221
determined experimentally by observing with it a beam in which
the percentage of polarized light may be regulated at will. This
may be accomplished either by setting a plate of glass at an angle
of 45°, and varying the relative intensities of the reflected and re-
fracted beams, or by reuniting two beams of variable intensity by
means of a double-image prism. Again, if the beam is strongly
polarized it is impossible to make fche bands disappear, unless a
large number of plates are used, in which case the transmitted
beam is very feeble.
These various difficulties are obviated by the other form of
polarimeter. As in the Arago, two adjacent images of the square
are formed, one polarized horizontally, the other vertically, which
will have equal intensities if the light is unpolarized, but one of
which will be in general, brighter than the other, when viewed by
polarized light. If now the two images are seen through a Nicol's
prism, their relative intensities will vary as it is turned, each dis-
appearing when the plane of the Nicol is perpendicular to its own.
Accordingly, certain positions can always be found, in which the
two images will have precisely the same brightness, and the angle
through which the Nicol has been turned, gives a measure of their
true relative intensities, and hence the percentage of polarized
light present. To make the reduction, call a the angle through
which the Nicol has been turned, A the amount of light polarized
in a vertical plane, and _Z? that polarized horizon-
tally. Thus if the plane is vertical, A is greater
than .Z?, and A — B is the amount of free polarized
light. A + J3 being the total intensity of the
light, their ratio gives the percentage of polariza-
tion. When the plane of the Nicol is vertical, A
retains its full brilliancy, which, at any other angle
is reduced in the ratio cos2 a. JS is in like manner
proportional to sin2 a. The percentage of polarized
A — B cos2 a — sin2 a
light n therefore equals —A — j — ^ = « ; — :— 5 —
A ~r -B cos2 a -f- sin2 a
= cos2 a — sin2 a = cos 2a. The reduction may then be effected
by a table of natural cosines, or by the accompanying table which
gives the corresponding values of a and n.
To use this instrument direct it towards the source of light to
0 100.0
5 98.5
10 94.0
15 86.6
20 76.6
25 64.3
30 50.0
35 34.2
40 17.4
4o 0
222 SACCHARIMETER.
be examined, and turn it so that the line separating the two
squares shall be parallel to the plane of polarization. Then turn
the Nicol until the two images are equally bright, when the angle
will give, by means of the table> the percentage of polarized light
present. It is best to take readings on each side of the 0° and
employ the mean, thus eliminating any error in the 0° point.
Otherwise, care must be taken that the circle is fixed in such a
position that when the Nicol is turned so that one image shall
completely disappear, the reading of the index shall be precisely 0°
or 90°.
Now measure with the polarimeters the amount of polarized
light contained in the rays whose plane of polarization was pre-
viously determined. Next throw a beam of sunlight upon a
sheet of paper, and measure the percentage of polarization of the
light thrown out in various directions. Observations of this kind
are much needed for various substances at different angles of
incidence and reflection. It will be found that it is extremely
difficult to obtain a beam from a large surface entirely free from
all traces of polarization, and hence much care is needed to ob-
tain really accurate results.
When the sky is clear its light is found to be strongly polarized
in planes passing through the sun, the effect being most marked,
at a distance of 90° from that body. Beyond 90° the polarization
again diminishes, and becomes zero at a point called the neutral
point in the same vertical plane as the sun, but 150° distant,
below this point the plane of polarization becomes horizontal.
Two other neutral points exist, one 17° below the sun, the other
8£° above it, -but both much more difficult to observe. Even a
faint cloud alters these effects, and when the sky is entirely
covered with clouds, no polarization is perceptible. Very valuable
work might be done by measuring the plane and amount of polar-
ization of the light in different parts of the sky.
94. SACCHARIMETER.
Apparatus. A Soleil saccharimeter, some pure sugar and some
unrefined, or brown sugar. Also some chlorhydric acid and sub-
acetate of lead, a flask containing just 100 cm.8, a funnel, filter
paper, a balance and weights.
SACCHARIMETER. 223
Experiment. The most important practical application of po-
larized light is to sacchariraetry, or the measurement of the
strength of a solution of sugar. This depends on the property of
such a solution of producing rotary polarization, or of turning the
plane of a beam of polarized light by an amount proportional to
the amount of sugar present. The saccharimeter is merely an
instrument for measuring the angular change of the plane, or more
strictly, the thickness of a plate of quartz, rotating it in the oppo-
site direction, required to bring it back to its primitive position.
The liquid is contained in a brass tube closed at each end with
plates of glass, which are held in place by screw caps. The light
first passes through a circular aperture, two polarized images of
which are formed by a double-image prism, and one transmitted
through the instrument, the other thrown off to one side. It
next passes through a double plate of quartz formed of two semi-
circles, one of which turns the ray to the right, the other to the
left. It next passes through the column of sugar by which both
rays are turned, by a certain amount, to the right. It is brought
back to its primitive position by a compensater, formed of a plate
and two wedges of quartz, the latter being turned in opposite
directions, and carrying racks which are acted on by a pinion so
that they may be moved past each other by any desired amount.
The thickness of the layer of quartz may thus be varied at will,
and accurately determined by a scale attached to one wedge, and
an index to the other. The light next passes through a Nicol's
prism which serves as an analyzer, and then through a small Gal-
ilean telescope by which an enlarged image of the biquartz is
formed. In front of the eye-piece of the telescope is an additional
Nicol's prism and plate of quartz, the latter being free to turn.
The object of this is to vary the tint of the two halves of the bi-
quartz, so that the color to which the eye is most sensitive may be
selected, and also to neutralize any color already present in the
solution.
To use this instrument, weigh out 16.47 grammes of pure sugar
and dissolve it in enough water to make the solution occupy 100
cm8. Unscrew the cap from one of the tubes, fill it with water
and slide on the glass plate, taking great care that no air-bubbles
are imprisoned under it. Replace the cap and wipe the exterior
224
SACCHARIMETER.
dry. Fill a second tube in the same manner with the solution of
sugar. Turn the stand towards the light, lay the tube containing
water in place on it, and focus the telescope on the biquartz by
drawing out the eye-piece until the line of separation is distinctly
visible. The two semicircles will now, in general, appear of differ-
ent colors which may be changed by moving the wedges by the
milled head below. A certain position will be found, however, in
which they are alike, and the reading of the scale should then be
zero.
When the two halves appear of the same tint, turn the quartz
plate in the eye-piece, by which their color is altered. There is a
peculiar purplish brown color, different for different eyes, from
which the two halves change more rapidly than from any other,
when the wedges are moved. Consequently, when of this
color, which is called the sensitive tint, they can be set more
accurately than in any other case. To obtain this sensitive tint
bring the two halves as nearly alike as possible, then turn the
quartz and see if any difference is perceptible ; if so, set again, un-
til no difference can be detected in the two halves, however the
the plate is turned. Take a number of observations, and reading
the scale to tenths of a division, take the mean. If not zero, it
may be brought to this position by means of a small screw, which
moves the scale without affecting the wedges.
Now replace the tube containing water by that containing a
solution of sugar, when it will be found that the semicircles
have very different colors, and on making them alike, the reading
of the scale becomes 100, if the sugar is perfectly pure. As before,
take the mean of several readings, and turn the quartz each time
to obtain the most sensitive tint.
Next make a solution of one half the strength by mixing some
of the standard solution with exactly its own volume of water, and
see if the reading is 50. Then dilute again one half, to get a
solution of strength one fourth, and see if the reading is 25. If
kept for some time, the solution will ferment, and the reading
diminish, especially during warm weather, or if exposed to the air.
In general, a solution of impure sugar is not transparent, and is
often so opaque, that the semicircles cannot be observed through
it. In this case it must be clarified by adding some sub-acetate of
SACCHARIMETER. 225
lead and then filtering. Animal charcoal was at one time used,
but it is found that this absorbs some of the sugar with the im-
purities. In practice the problem generally is complicated by the
fact, that the molasses and other impurities commonly found in
sugar, also turn the plane of polarization to the right, and thus
render the results uncertain. This effect must therefore be elimi-
nated by adding to the solution one tenth of its bulk of pure
chlorhydric acid, and heating to 68° C. The cane-sugar is thus
converted into grape-sugar, which turns the plane of polarization
by an equal amount to the left, while it does not effect the mo-
lasses and uncrystallizable sugar. After heating, the solution is
poured into the larger tube, which has an aperture in one side to
contain a thermometer. The length of the column in this case
is one tenth greater than before, which just compensates for the
dilution due to the addition of the chlorhydric acid. The reading
should then be taken, and the temperature noted. As this read-
ing gives the difference of the amount of crystallizable and un-
crystallizable sugar, and the first reading gives their sum, the
amount of crystallizable sugar may be obtained by taking half the
sum of the two readings, and the amount of uncrystallizable, by
taking half their difference. Thus if this first reading is 80 and
the second 30, it denotes that there is 55 per cent, of crystalliza-
ble, and 25 per cent of syrup. A correction must be applied for
temperature, which is best done by means of a table, which ac-
companies the instrument.
15
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